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ii'ni^t)M>!'!i^ A''
ttlTICS.
%
SCIENTIFIC
AMUSEMENTS
IN
PHILOSOPHY AND MATHEMATICS.
Planner and Brewis, PrinterSy
^ Lore LuiMt little Eutchcap.
. L
>
SrUUXT IVI'- A.V! L'SE^ITii^rTS.
SCIENTIFIC
AMUSEMENTS
IN
PHILOSOPHY AND MATHEMATICS
INCLUDING
ARITHMETIC^
ACOUSTICS^
ELECTRICITY^
MAGNETISM^
OPTICS,
PNEUMATICS,
TOGETHER WITH
AMUSING SECRETS
IN VARIOUS BRANCHES OF SCIENCE,
The Whole calculated to form
AN AGREEABLE AND IMPROVING EXERCISE FOR
THE MIND.
Particiilarly recommended as
A USEFUL SCHOOL BOOK.
BY
W. ENFIELD, M. A.
Author 9/Naimni Iftrolofft Tk« Yowig ArtUt'i JtiUtamt, Promomtetng Dictimarp, 4r.
LONDON :
PRINTED FOR A. K. NEWMAN AND CO. 8IMPKIN AND MARSHALL,
T. TEGG, AND EDWARDS AND KNIBBS; ALSO GRIFFIN
AND CO. GLASGOW.
1821.
■■\ r-
sr^^ .\
^H
>■
iS .-.Qi'S
CONTENT S.
Page.
PREFACE t
INTRODUCTION 7
Of Fractions ..•...•*••• • ib.
Of Powers •....• 9
Of Equations ••••• •••* 12
General R ales in regard to Equations •• ib.
Of Ratios and Proportions • • • • ......# ib.
Properties of Arithmetical Proportion and Progression 14
Properties of Geometrical Proportion and Progression 17
Ruleof Ihree 19
ARITHMETIC - SO
Of our Numerical System, and the different Kinds of
Arithmetic ib.
Of some Properties of Numbers S3
Of Arithmetical and Geometrical J'rogression, with .
some Problems which depend on them ••<••• ib.
Of Harmonical Progression 32
Problems in Geometrical Progression 33
Exercises in the Single and Compound Rule of Three, ^
both Direct and Inverse ^••••a 38
Single Rule of Three Direct •••• • ••• ib.
Single Rule of Three Inverse 39
Compound Rule ot Three • • •' • • • • ib.
Rule of Fellowship •• •••..... • 40
Rule of Alligation • . . . . .^ . • 42
A Chronological Problem ••••• 45
The RuleofTare - 46
Of Combinations and Permutations ••••••• 47
Problems in Combinations and Permutations 50
Application of the Doctrine of Combinations to Games
of Chance and Probabilities .••••• •• 55
Problems in Probabilities and Games of Chance 56
A I'able of the difi'ercnt Ways in which any Point can
be thrown with One, Two, Three, or more
Dice 61
Arithmetical Amusements in Divination and Combi-
nation • 72
Different Methods of telling what Number a Person has
thoughtof 73
A Person having in one Hand a Piece of Gold, and in
the other a Piece of Silver, to tell, &cc 78
The Game of the Ring 79
To guess the number of spots on any catd \«>q\c\\ ^
' person has drawn trom a ¥rho\e p«Lck« •«%•%%% ^^
VI CONTENTS.
Page.
A person baTing a certain number of counters in eaoh
hand, to tell bow many he has altogether* • • • 8t
Several cards being given, to tell which of them a per-
son has thought of 82
Another problem of the same kind ••••• ib.
To make all the cards of the same kind be found toge-
ther, however often the pack may have been
cut * 83
The four indivisible kings • • • ib.
To write down on a piece of paper the heap of cards
which a person will choose 84
Several cards being presented to several persons, to
guess that which each has thought of ib.
Three cards being presented to three persons, to guess
that which each has chosen 84
To tell the number of spots on all the bottom cards of
several heaps, arranged on a table 86
To name all the cards of a pack* •- ib*
To make a person believe you can distinguish the
cards by their smell 83
A pack of cards being divided into two parts, to disco-
ver whetehr the number in each be odd or
even ..• ib.
To tell rhe number of spots on several cards which any
person has chosen* • • • ib.
A person having drawn four cards from a pack, to tell
the one he has thought of 89
Three things privately distributed to three persons, to
guess that which each has got ib.
To tell^ by inspecting a watch, at wh^t hour a person
has resolved to rise next morning 91
Two persons agree to take alternately numbers less
than a given number> and to add them toge-
ther till one of them has reached a certain
sum, by what means one of them can reach
that sum before the other • • • • • ib.
Sixteen counters being disposed in two rows, to find
that which a person has thought of 92
A certain number of cards being shown to a person, to
guess that which he has thought of 93
To arrange 30 criminals in such a manner as to save
15 of them, &c • • • 94
The game of the nosegays ib.
A man has a woli; a goat, and a cabbage, to carry over
a river, &c •••••••••• 95
To dispose counters in the eight external cells of a
square, so that there shall always be nine in
each row, and yet the whole number shaU
vary from 20 to 32 • ••••• ib
k »
CONTENTS. Vll
Page.
To distribute among three persons 21 casks of wine> 7
of them full, 7 of them empty, and 7 of them
half full, so that each of them shall have the
same quantity of wine, and the same number
of casks • • 97
^. schoolmaster, to amuse his scholars, &c. ib.
To tell the figure which has been privately cut ofif from
a certain product • • • • » 98
A person having multiplied two numbers together, to
tell the product, provided you know only the
last figure of it 99
A person having chosen two numbers, and divided the
greater by the less, to tell the quotient ib.
POLITICAL ARITHMETIC 101
Of the proportion between the males and females • • ib.
Of the mortality of the human race, according to the
different ages • 102
Of the number of men of different ages in a given
nuniber • 105
Of the proportion of the births and deaths to the whole
nuniber of the inhabitants of a country 106
Of some othef proportions in regard to the inhabitants
of a country * 107
The age of a man being given, to tell the probability of
his reaching a certain age • • • • 108
A young man, aged 20, borrows £1000. 1o be paid with
interest, when he attains to the age of 25, &c. 109
A slate or an individual, having occasion to borrow a
sum of money on an annuity. Sec • • • ib.
MAGIC SQUARES Ill
Method of constructing an Odd Square 112
Method of constructing an Even Square ib.
A Geometrical Square* •••- 113
To make the knight pass overall the squares of the
chess board, without passing twice over the
same • 114
APPLICATION OF ANALYSIS to the Solution of •
Various Problems 116
A lady lamenting that her age was triple that of her
daughter, &c. • • • • ib^
A father on his his death-bed, gaTe orders in his wili,&c. 1 17
A captain being asked how many soldiers he had in his
company »&c •• • ••'• lis
The head of a fish nine inches in length, *&c. 1 19
A person who had the lease of a house for 99 years,
being asked, &c. • ib.
To divide the number 50 into two such parts^&c « \^<C^
It is proposed to divido 100 into two sucV\ Y>a\U, ^c. • • \>a.
Two persons sat down io play, &c. ••• \'>.V
VIU CONTENTS.
Page.
The minute band of a clock being at 12, and the boar
hand at 1, &c. • • • • • • 122
If two bodies move towards each olher with unequal
▼elocities, &c ••••••• ib.
To divide 90 into two parts, the ratio of which shall be
as2to3 ^••« ' US
Application of Analysis to the solution of the I'lth pro-
blem of divining arithmetic • • ••••-•••••• ib.
What number is that the f of } of which, &c. 134
What number is that the i of f of which, &c. ••..•• • ib.
What number is that f. of | of which, &c. •-•• 125
What number is that of which f of J, &c. • • • ib.
What number is that of which J + i are equal to 1 ? • • 126
What number is that the i, {, and i of which make 12 ib.
The triple, the half, and the fourth, &c. ib.
If i and I of the hull of a ship, &c 127
A banker at his death being desirous of rewarding ten
of his clerks, &c. ib.
TABLES OF CHANCES on games of play 130
ACOUSTICS AND MUSIC 143
Definition of sound, &c • ib.
Of the velocity of sound— method of measuring dis-
tances by it • • • 145
How sounds may be propagated in every direction
without confusion 146
Of echoes— how produced — account of the most re-
markable, and of some phenomena respecting
,them - • ib.
To construct two figures, to be placed at the two ends
of a hall, one of .which shall repeat to the ear of
a person, &c. •••• ••• 149
Experiments respecting the vibration of Musical
Strings ••• • 151
Scale of Sounds in the Diatonic Progression • 152
To determine the vibrations made by a string of a
given length and size •••• • 154
Method of ad.ding, subtracting, multiplying, and divid-
ing Concords • • • « * • 156
To add one concord to another ...•••#•••.• ib.
To subtract one concord from another* • • • • 158
To' double a concord, or multiply it any number of
times, at pleasure • ib.
To divide one concord by any number at pleasure" • • 159
Of the resonance of sonorous bodies, the fundamental
])rinciples of harmony and melody, with some
other liarmonical phenomena ! "." * ^^^
On the harmonical sounds hetffd with the principal
sound 162
Of the modem music #..#.... ,..,.♦• 163
CONTENTS. IX
■ «, ^'^•
inecuiM of the pleasure wiWDgfrom mnsio 1«3
Oflhe properties -of certsin iiislrumecls, kc. J69
Of lome musical inBtnitoenIs or macliiDes remarkable
for Iheit singiilarilj orcoDitruclion 171
O/anewiiiBlrunicnieaDeillLe hannonicB 172
On what is called 4 false voice 173
Of the speaking trumpet and earlrnmpel 175
ELECTBICITY 178
Definitions ■ ib.
ApboriBniB - 179
Esperimeuts in electricity 181
The animated fealber 184
The artificial spider ih.
Tbe marvellous fouutain 185
The Magic Pichire ib.
The TautaJian cup 186
The Belf-moTiQg wheel • ib.
The magician'.! tbace J88
The plaueiarium 189
The incendiaries ib.
Tlie iDcouceivable shock 190
Magieal explosions ■ 193
The prismatic colours ib.
The artificial spider 103
The artificial earthquake 194
The electrical kite jb.
Candle lighted by eleotricitj 1D3
Candle bombs •' .■■. ib.
Dancing balls ........ ,.... ib.
TheLojden phial 195
BosId ignited b; electricity 197
Spirits iDiiiiied by electricity ib.
£lectrihed air ib.
To spin -'iealing-wax into threads by eleclrleity .... 1 98
£lcclrJlied camjilior 1!I9
Electrical amusements in the dark chamber ib.
The fiery sbower ib.
Uiraculous luminaries 20U
The globuliirfifCB 301
Qlie illuminated vacuum 202
The JamiiiouB cylinder 203
The magiciil cun«<ellalians ■, ib.
The aurora liorealjs 204
Circulating lampa .h 305
MAGNETISM 306
Definitions ib.
Aphorisms 2or
The magnetic wand 903
X CONTEim.
Page.
Thcmj«(eriouiwatcL "• WW
Tlie magnetic dial • ib.
Tlio masiietic cards 90^
TJie comiDunicaliTe ciown • 210
TJie magnetic table ; Sll
The iucDioprehenaible card 813
PNEUMATICS 213
Definitions ib.
Apbomms ib.
Tfie bottle broke by air 314
The brass betuisphere 21S
■Water boiled bv air ib.
The ieiial bubbles 916
The flouting alone Jb.
The witlitrud fruit restored ib.
The Tegetable air bubble ib.
'Jhe mereariai rod 217
The iiiyslicalbell • ib.
Fcalbcrs heavier than lead 218
The solf-inovioK wheel ......•■-•■ ■.•• ib.
The animated ligure 2l»
The arlifitial halo ib.
The tdcreuripl shower OTO
The fouDlaiu in vacuo * ib.
The Rcmciited bladder ib.
Cork hnaTierlhan lead >b.
The aiiimalcd Bacchns 831
The arlifioial balloon ib.
£K|icrimrtit with a viper *••• ib.
ExpeiinieDla with Sparrows ....< 22g
OPTICS 2M
DefinitiuDs ib.
Three objectBdiacerniblo only with boUi eyes 328
To conslriicl the camera obscura S29
Tlie magnifying reflector ••• SSO
Optical Augmentation ib.
To magnify small objecli by means of the son's rajs let
into a dark cliamber--.--. .-..• 331
The m agio lantern ib.
Method of painting the glasses for the lantern 334
AMUSING SECRETS S86
To make a ring be snspended by a thread after it bai
been burnt ■■-• 'b.
To make people in a room have a hideous appear-
To fonn flgurei in relief on sn egg ■.•• ib.
Page.
IVuhsnge a colour fronr white to blue 296
IWtifDLe a red Kqnor, which, when poared ittto diflbr-
ent glasses, shall become yellow, blue, Mack,^ at
purple^ ••• 237
To Make porfiatanr with water anxl wa« • • • • ib.
Haw a body of comfonstrble matter may be penetrated
by fire without berng consumed • • • • ib,
Apprent transmutation of iron into co^yper or sliver • • 238
Dirorent substances successively precipitated by add*
ing another to the solution • • • • 239
By thenrixtore of two transparent licj^ndrs to prodtice a
blackish liquor — method of making good ink* • 240
To prothfce- inflammable and fulminating vapours* • • • 241
Tk^" philosophical candle • • • • • • ib.
To mak« an artificial volcano ••.....*• 24t
To make fulminating powder- • » • • • • ib.
To form a combination, which when coM is liquf'd and
transparent, but when warm becomes thick and
opake • •• 245^
To make a flash, like that of lightning, appear in a
room when any one enters it with alighted candle ib.
Of sympathetic inks, and some tricks whicb may be
performed by means of them ••••••• S44'
To make a drawing, which shall alternately represent
winter and summer •• ••..•• •••••• 245
Tfa*" magic oracle •••• ••• 245
Of Itt^tallic vegetations ••••••••..•••• •...• jb.
Arbor Martis, or tree of Mars 347
Arbor Dianae, or tree of Diana •..•..•...• ib,
Tbeleadtree 248
Koii-metallic vegetation** • • •, • ib,
Tai produce beat, and even flame, by means of two cold
liquors • • 240
Toifilse-iron io a moment, and make it mn into drops lb.
Cement for making broken china* •••••• 250
Prtyoess for whitening prints • • • ib;
MVItlKMi of taking paintings from the old canvass, and
transferring' them to new • ••••••••••••• 251
To fill a glass witii water in such a manner that a per-
son shall not be able to remove it without
spHling'it all qs2
Toeonttnict two figures, one of which shall blowout
« candlo> and the other light it again -^o • ib«
Ji^sn vases ••••••»•• • ib,
To^oomtmet a vessel iVom which water shall escape
through the bottom, as soon as its mouth isun«
stopped .-..t 259
Xn CONTENTS.
Page,
Transparenciei • 353 ■
Method of fixing crayoDi ' ib.
A coriona illusion 367
Ab object being placed behind a convex glass, to make
it appear Wore it ib.
The Chinese ahadowa ib.
To direct a swarra of bees at pleasuie 259
A ponder which intlamea when exposed to Ihe ur.*" 260
Fulmiualing gold • ib.
To cut glass by means of heat •••■ 261
To melt a piece of money in a walnut-shell, without
injuring the shell ib.
Pboaphoms 262
A liquor which shines in the dark •• ib.
To malfc luminous characters appear on a piece of
paper, or a wall, &c 283
A liquor shut up in a bottle, which when the botUe is
nnstopped, becomes luminous ib.
Method of speedily delineating all sorts of plants and
flowers • ib.
llie cliangeable rose 264
The magic picture ib.
The cbaugeable picture • 265
Golden ink • ib.
Process translated from the French 906
White ink, to write on black pq>er • ib.
Red ink «67
Blaeink ■ ih.
Yellow ink ib.
Green ink "b.
Ink of difi'erent colours, made from the juice of tiolets ib.
Tracing Ink 268
China or Indian ink ~ 269
Ink powder S70
To reviye old writing ib.
To take off the impression of any drawing 271
To take off the impression of old prints 272
Method of leaching drawing to young persoDS - ib.
To coualruct a lantern which will enable a person to
read by night at a great distance ib.
To take off impresalons in plaster of Paris or sulphur- 373
Baits for catching fish ib.
To produce Tariely in the colours of Bowers ••..• 975
To obtain double flowers ■•■ ib. '
To obtain flowers of different colours on Ifae hudo stem ib.
PREFACE.
TO trace out the origin of amusements, it appears
that it would be necessary to go back to the earliest
ages of the world For mankind, being exposed to a
rariety of fatiguing labours, which exhaust both the
mind and the body, have at all times exercised their
ingenuity in devising means to dispel melancholy^ and
to revive the depressed' spirits.
The remedies pointed out by nature for this purpose
are, rest, proper nourishment, and cheerfulness : each
day indeed exhibits in the same individual a new being,
in good or bad spirits, according to the impressions
made on the animal economy, by rest, a change of
food, and various other circumstances.
The mind is too intimately connected with the body
fiot to participate in the evils by which it is affected;
but to the former, rest alone is not sufficient: to revive
its powers, and to exhilarate the spirits by a proper
stimulus, a change of objects, amusing conversation,
agreeable news, and other things of the like kind, are
necessary^
Every one knows, that the spirits are depressed by
too long application to gloomy or serious objects : te
remedy this evil, others more amusing must be sub-
stituted in their stead ; the least trifle or toy k often
2 PREFACE.
capable of giving to the mind the most tranquil and
agreeable impressions; and during this state of peace
and repose, new spirits are created, which produce a
change in the whole frame.
Walking, hunting, dancing, and music, are excellent
sources of amusement ; but they are not the only ones
to which the necessity of unbending the mind, and fill-
ing up a vacant hour, have given birth.
The game of chess, it is said, had its origin at the
siege of Troy; being invented by Palamedes, to amuse
the Grecian chiefs, disgusted with the tediousness of
the siege.
Cards and tennis were invented by the Lydians, a
people of Asia Minor, among whom, according to the
antiquarians, all games had their origin : these people,
it is well known, were so much addicted to voluptuous-
ness and gaiety, that to express a thoughtless, careless
* action, it was said, proverbially, to have been done
Lt/diomore.
Amusements, then, are remedies invented to revivo^
the depressed spirits, and to render the mind capable
of resuming its usual labours with greater success ;
but a wise man will employ them with moderation, and
will consider them as objects calculated to unbend the
mind, and not to occupy it entirely.
Cicero told his son, that amusements ought to be
employed like sleep ; which, if used to excess, becomes
dangerous, and instead of reviving the powers of the
mind, renders them torpid.
On this subject, Cassian relates an expression of the
Apostle John, which deserves to be recorded. — A hun-
ter, who one day saw him caressing a partridge, seemed
astonished that so pious a man should amuse himself
with such a trifling object. " My good friend," said
PREFACE. 3
the apostle, " what have you got in your hand?" —
" A bow," replied the hunter. " And why is it not
bent?" added the apostle. — " If it were always bent,*^
returned the hunter, " it would lose its strenffth."
** Be not then surprised," continued the apostle, " that
the mind also should sometimes require relaxation."
Sidronius Hoschius, the Flemish Ovid, has expressed
the same thought, with great elegance, in the following
lines:
*' Deficiet sensiro qui semper tendltar arcus ;
Ferre iiegat segetes irrequietus ager.''
The atter comparison has been employed by Seneca,
who says, " Tiie mind of man is like those fields, the
fertility of which depends on their being allowed certain
periods of rest, at the proper seasons." This philoso-
pher had remarked, that too long and too assidious la-
bour exhausts the mind, throwing it into a kind of lan-
guor; but, that by relaxation it is revived, and ren-
dered fitter for resuming its occupations.
How often are people difficulted by problems merely
of an amusing nature, the whole solution of which de-
pends upon some elementary calculation, the natural
properties of certain bodies, or mathematical combina-
tions! We admire the sagacity and pretended know-
ledge of the person who proposes them; and yet no-
thing is easier than to comprehend ^ and even to exe-
cute, what thus excites our astonishment and wonder.
Why then should not we acquire the knowledge neces-
sary to enable us to propose problems and enigmas
ourselves ?
Intricate and puzzling questions have, at all times,
formed a part of the amusements of the most polished
nations; and they have been received with avidity,
b2
% PREFACE.
•Ten by young persons, when presented under the
agreeable form of an enigma or recreation. We may even
venture to assert, if we are allowed to judge of others
by what we experience ourselves, that we are some-
times conducted to the higher parts of the most ab-^
stract studies, by the flowery path of some etperiment,
Hfhich we at first considered as an object of mere cu-
riosity.
It is well known, that the high reputation of Solo-
mon induced the queen of Sheba to come from the re-
motest part of Ethiopia, to admire the wisdom of that
^eat prince — the wonder of his age. She came, says
th^ scripture, to try his wisdom, by proposing to him
enigmas. Solomon satisfied her on every point, and
answered all her questions with so much propriety, that
the queen returned in the utmost joy, unable to con-
tain the transports of her admiration, excited by th«
wisdomi and magnificence of that great king.
The celebrated ^sop became the favourite of
Croesus, merely on account of his ingenious fables,
^ which contaiped the most refined morality, and in-
•tructions the more delicate, as they conveyed cen-
sure without wounding that self-love which is so natu^
ral to man. Fable, in the hands of this great genius,
seemed a rod dipped with so much art in the gall of
satire, as to have none of its bitterness or severity.
Kathan represented to David the enormity and injustice
ef the crime he had committed, in regard to Uriah,
only under the veil of an ingenious allegory ; which
produced a greater and speedier effect on the mind of
the monarch, than if the prophet, arming himself with
the thunder of his eloquence, had pointed out to him,
in a direct manner, the horror of his offence.
Does not Solomon desire the sluggard and the
PREFACE. 5
spendthrift to consider the ways of the ant; and does
not iBsop seem to have borrowed from this idea hit
fable of the ant and the grasshopper? That great
prince in delineating the portrait of true wisdom, paints
her in saying, '' To understand a parable, and the in-
terpretation, the words of the wise, and their dark
sayings."
Mental amusements, then, have been esteemed, in
all ages, and by persons of every condition ; and the
pleasure they excite is the purer as they affect only the
more delicate parts of the mind. The human intellect,
as is well known, has its peculiar pleasures; every
thing that increases knowledge, pleases and exalts it;
we are always gratified when we comprehend a diffi-
culty whiph has checked the progress of others, or have
unveiled a mystery, concealed from persons possessed
of less penetration than ourselves.
Besides, these amusements, purely intellectual, ^lay
be enjoyed at little expence ; they do not fatigue the
body; on which they make no impression ; and, on this
account, they ought to be preferred to sensual plea-
sures, the enjoyment of which creates disgust, ii^ures
tlie health as well as fortune, and almost always de-
ranges the economy of a peaceful and tranquil life.
The class of mathematicians has always arrogated
the right of treating of mathematical and philosophical
recreations. In compiling the present collection,
Ozanam, and those who have written on the same sub-
jecthave been our guides; and from their works we have
selected the greater part of what we now offer to the
public ; Jot these amusements are the production nei-
ther of one man, nor one age, but of a great number
of the learned, of artists, and. of many ages of researck
and of observation.
Bd
O PREFACE.
By the long experience we have had, we are In-'
duced to hope that young persons, who are often dis-
gusted with the formaUty of study, and who, on that
account, sometimes conceive an aversion to the most
useful branches of science, will find in the greater
part of the amusements which are here presented to
them, some things suited to their taste, and easy to
be comprehended. When the first difficulties are sur-
mounted, they will become so many steps to conduct
them gradually, by the most agreeable path, to the
solution of problems, which at first may appear too
difficult and abstruse for their age.
As it is impossible to understand properly all these
amusements without the knowledge of certain princi-
ples, the application of which is often necessary, they
are preceded by an introduction, calculated to facili-
tate the solution of the most difficult problems.
#*# The reader is requested to observe, that the
figures, inclosed within parenthesis, which occur in
the course of the following work, refer to that section
in the introduction, where the necessary explanation
will be found.
AMUSEMENTS.
9
INTRODUCTION.
1. DIFFERENT symbols or signs, established by
general practijcey are sometimes employed in order to
simplify calculations, and facilitate the resolution of
certain problems. Thus,
4- signifies plus or inOr6.
— minus or less.
=; •••••• equal.
> •• greater.
< less.
X •••••• multiplied by.
-i. divided by.
Thus, it may be easily conceived that 2 + 3 =:- 5 ;
that 3 — 2 =: 1 ; that 3 > 2; that 2 <{ 3 ; and that
12 ^ 3 or y« = 4.
OF FRACTIONS.
2 . Besides the application of the common rules to
whole numbers, with which every body is acquainted,
it is sometimes indispensably necessary to perform th«
same operations with fractional numbers.
A Fraction is one or more parts of a whole. Every
fraction is expressed by two characters, placed one
above the other, with a line between ^em, in this
manner: -, ~ &c. The upper character, whicli \%
8 AMUSEMENTS
called the numerator, expresses how many parts arc
taken of the whole; and the other, called the denomi->
nator, denotes the quality of these parts. Thus the
fraction ^ signifies that the whole is divided into
fourths, and that 3 of them are taken.
It hence follows, that a fraction is greater according
as the numerator is greater ; and, on the other hand,
less as the denominator is greater. Thus ^ > ^;
^ < ^. By a necessary consequence, all fractions, the
two characters of which are equal, denote exactly the
same value; f = J = ^ iz |, &c.
3. To reduce a whole number to ^ fraction, which
shall have a determinate denominator, we must mul-
tiply the whole number by the given denominator, and
place the product above the latter. Thus 8, reduced
to a fraction, having n for its denominator, is ^ : and
5 reduced to a fraction having the same denominator
as 4, is y.
4. To reduce two fractions to the same denominator,
the numerator of the first must be multiplied by the
denominator of the second, and the numera1;pr of the
second by the denominator of the first : these two pro-
ducts will be the numerator of two new fractions; and
the product of the two denominators will be the com-
mon denominator. Thus |. and |-, reduced to the same
denominator, give 4? and •^. Any number of fractions
may, in like manner, be reduced to a common denomi-
nator, provided that each numerator be multiplied by
the denominators of the other fractions, and that tlie
product of all the denominators be taken for a common
denominator. Thus, for example, the three fractions,
h T' y* when reduced to the same^ denominator,
8^^^ -84—
5. To add two fractions, we must first reduce them
to a common denominator, and then add their nume-
> rators. Thus the sum of the two fractions <| and ^, is
—35 ft-
6. To subtract one fraction from another, they must
first be fedttced to the same denominator, and uie nu*
INTRODUCTION*
(.
merator of the less must then be taken froiin the ivu*
merator of the greater. Hence the difference of the
fractions -f and j is -^^
7. To multiply two fractions together, we must
make a new fraction, the numerator of which shall be
the product of the two numerators, and the denomina-
tor. Thus the product of f- by | is t\« . \
8. To divide one fraction by another,, we must i^ake
a fraction, the numerator of which shall be equal to
the product of the numerator of the first multipUed by
the denominator of the second; and the denomitiatoir
equal to the product of the numerator pf the . ^econd
multiplied by the denominator of the first. The quo«*
tient of ^ divided by -J, will therefore be f.
9* Sometimes it is necessary to simplify a fraction,
by reducing it to its simptiB^t expression, or what is
called its lowest terms : nothing js necessary for ^his
purpose, but to divide the numerator sind denominator
by the greatest common measiire* or divisbr., Thus the
fractions | and 7^, reduced to th^ir simplest expTdi"
sion, give ^ and i» \
OF POWERS,
10. By the power of a quantity, is understood its
product by unity or by itself a certain number of time*.
Thus, the first power of 2 is 2 : its second power or
square is 2 x 2; its third power or cube is 2 x 2 x 2,
and so on. .Hence it is evident, that to obtain any
power whatever of a given quantity, it must be multi-
plied by itself as many times less 1 , as are equal io
the number which denotes that power.
The |)ower of any quantity is expressed sometim'es »
algebraically, or numerically,- by the fi^re which de-
notes its degree, as in the following examples: a^,' d*^
a», «♦, &c. 4«, 4», 4», A^i &c.
11. An algebraic dharacter it sometimes accompa*
nied by two figures, as 26* . The first of th^se is ^ed
the coifflcitviy and the secoild the expmtnt: the former
denotes how many times the <panthy is added tA
itself; and the second indicates the po^ei« TW%)^%
25
10 AMUSEMENTS.
value of by being supposed equal to 3, we shall Imve
26» = 54.
1% By the root of a quantity, is meant a number
which being multiplied one or more times by itself, will
give that quantity. There is therefore a square root
cti6e rooty &c.
The different roots of quantities may be expressed
by the following signs: ^, ^, \/9 in this manner,
^ay l/Gy t/ay ^16, 4/27.
13. We shall now shew how to extract the square
root of any quantity ; that is to say, how to find a
number, which being multiplied by itself, will give
that quantity, if it be a complete square, or at least the
greatest square which it contains.
EXAATPLE I.
r Let the number, the square root of which is required
be 1156. First divide this number from right to left
into periods of two figures, and then proceed as
follows :
Find the greatest square contained in n 55 (34
11,. which is 9, and write down its root g'
3, as seen in the annexed example. Square r>^
3, which gives 9, subtract it from 11, and .
the remainder will be 2. Then bring down ___
the next period, which is 5Qy that it may
serve as a dividend along with the figure 2 on its left.
Take 6 as a divisor, that is to say, the double of the
root 3 already found ; place it on the left, and find
how often it is contained in 25 ; the quotient will be 4,
which must be written down in the root after 3; and
also after 6, the divisor, which will give 64. Then
multiply the last number by the second root 4, and
the product will be 256. As there is no remainder, it
is a proof that 1155 is a perfect square, the root of
which is 34. Had the last product been too large to be
subtracted, it would have been necessary to diminish
the last figure in the root, in order to make it small
enough for that purpose.
256
256
INTRODUCTION. 1 1
SXAHPLE II.
What is the Square Root of the Number 214369 ?
As in the preceding examples, we must first divide
the number into periods of two, from right to left, and
there will be as many figures in the root as there are
periods.
Then, as the greatest 21^ 43, 69 (463
square contained in 21 is 16, iq
the square root of which is 4, gg^ — 543"
write down the 4 in the root ; g' rig
square 4, which will give 16,
and having , subtracted it ^^;?^ tr^^
from 21, the remainder will '^ ^^^^
be 5. Bring down the following period 4 J, which, with
the preceding figure, must be divided by the double of
the root already found, that is to say, by 8. The quo-
tient of 54 divided by 8 is 6, which must be placed
after the first root 4, and also after 8 the divisor ; then
multiply 86 by 6, and subtract the product 516 from
<543. Place the remainder 27 under 516, and bring
down the next period 69. Take, as the divisor of 276,
the double of the two roots already found, which is 92.
Divide 27 by 9, and place the quotient 3 in the root
after 46, and also after 92. Then multiply 923 by 3,
and if the product 2769 be subtracted from the number
2769, there will be no remainder. The truth of this
operation may be proved by squaring 463 ; that is to
say^ by multiplying it by itself.
After the last subtraction, if any thing remains, it is
aproof that, though the root found is not exactly the
real root, it does not want unity to be so; but if it
were required to approach still nearer to the real root,
nothing would be necessary but to reduce the remain-
der to decimals, and to continue the operation, taking
care to separate the whole numbers in the root from
the decimals. However, as we propose here only to
give a few amusing problems, there will be no necessity
•for carrying the extraction of the square root beyond
whole numbers.
1^ AHITSBMBHTB
OF EQUATIONS.
14. As certain questions cannot be easily tesolved
without some knowledge of analysis and equations,
we shall here give a short explanation of them, and
such as may be easily understood.
By equations is meant the application of numerical
and algebraic rules, to the solution of different ques-
tions, which may be proposed respecting quantity.
The first and most difficult thing in analysis, is to
comprehend properly the state of the question, and
the relation which the known quantities bear to the
unknown, in order that they may be clearly expressed
in an equation. '
Every equation is composed of two members, sepa-
rated by the sign :=; and each member may con-
sist of several terms. An example of the whole may
be seen in the following equations '«-
7 = 34-4;^-5 = 2 + 1; 3 x 4= 12; |L=: 3.
There may be equations also consisting of algebraic
quantities alone, or in which arithmetical quantities
are mixed with algebraic ones, as in the following :
ar4-6=rfl;x-— jfzia-l-^; 3a— 5=: 4c — 2 jr.
GENERAL RULES IN REGARD TO EQUATIONS.
RULE I.
i
15. Any quantity may be transp6sed from one mem-
ber of an equation to another, without deranging the
equation, provided that the signs be changed.
Thus, as 12 — 3 = 9, Wfe may write 12 = 9-^3.
For the same reason \( a ^ x -|- 36 r= rf ^ y, we shall
have a-f5( — rf = * — 36.
This method of operation, in regard to equations, is
called transposition, and i» employed when it is nieces-
tnrj to free one member of an equation from any qn^Hf-
iky connected with it either by addition or ftttbime^
tion.
nmioDucftON. 13
IIULE IT.
16. "Wheta an unknown quantity is involyeiJ ih ih
equation either by multiplication or division, it may t>6
disengaged from it, in the first case, by division; ^d
in the second by milltiplication. For example,
b y
If 3x = b, then jp = -; and if ^ = a, then jr =: 3a.
o o
This method of disengaging an unknown quantity
will be more easily comprehended, if we give de'tenni-
nate values to the quantities a and b. If we suppose,
for example, that 6=12, and a = 8, the two equations
above mentioned will be reduced to the following,
x=z V* = 4, if = 24.
17. It appears, therefore, that the whole art of ana-
lysis consists, first, in comparing in the equations the
unknown with the known qu^titi^s, and disengaging^
them from each other by the means already pointed out,
in such a manner, that the known quantity may renmin
alone in one member of the equation, and the unknown
in the other.
To facilitate the solution of algebraic questions, the
unknown quantities are generally denoted by some of
the last letters of the alphabet, v, j^, z ; and the known
quantities by some of tne first, as a, 6, c, &c.
OF RATIOS AND PROPORTIONS.
18. Relation or Ratio is what results from the com*
parison of two quantities. As two quantities may be
compared with each other two ways, ratio is distin-
guished into two kinds, arithmetical and geometrical.
Arithmetical relation^ is that of two quantities com-
pared with each other by subtraction.
Geometrical relation^ is that of two quantities com-
pared with each other by^ division.
Thus, for example, the arithrtietical ratio of 12 to 4
it d; and the geometrical ratio of the same <i[tiiiDtilieSy
k 3 ; for 1« - 4 =s a, and 'i* =; 3.
14 AMUSEMENTS.
19. Proportion it an equality of ratios. As there
are two kinds of ratio, there are also two kinds of pro*
portion, arithmetical and geometrical: the first con-
sists in an equality of differences; and the second in
an equality of quotients.
"fevery ratio is expressed by two terms; the first of
which is called the antecedent, and the second the con-
sequent.
Two equal ratios forrii a proportion; which is either
arithmetical or geometrical, according as they contain
eit)ier the same difference or the same quotient. Thus
3 • 5 • . • 7 • 9, expresses an arithmetical proportion ;
the meaning of which is, that 3 is arithmetically to 5,
as 7 is to 9; and 6:3:: 16 : 8, expresses a geome-
trical proportion; the meaning of which is, that 6 is
geometrically to 3 as 1 6 is to 8.
The first and last terms of each of these proportions
are called the extremes; and the other two the means.
20. Proportion is con/tnuecf when the same term is the
consequent of that which precedes it, and the antece-
dent of that which fdltows it. Thus the two following
prc^portions, one of which is arithmetical, and the other
geometrical, are continued, yiz. -rr 3 • 5 • 7 ; "H- 4 : 8
: 16. The meaning of which is, 3 • 5 •.• 5 • 7;
4: 8:: 8: 16.
When a continued proportion has more than three
terms, it is called a progression. Thus -i- 1 . 3 . 5 . 7 .
9 is an arithmetical progression, and -H- 4:8: 16 :
32 : 64 is a geometrical progression.
I
Properties of Arithmetical Proportion and Progression.
' THEOREM I.
21. In every arithmetical proposition, the sum of the
extremes is equal to that of the means.
If 3 . 5 •.• 7 . 9, or 9 . 6 V 8 . 5.
. Then 3 + 9 = 64-7, and 9 + 5 = 6+ 8.
It hence follows, that when three terms of an arith-
metical proportion are known, we may easily find the
INTRODUCTION. 15
fourth ; for if the unknown term be an extreme, h will
be found by subtracting the other extreme from the
sum of the means ; and if it be one of the means, by
subtracting the other mean from the sum of the ex«
tremes.
If a • 6 • . • c • 4r, or if 4 • a: • . • 3 • 8,
It hence follows also, that if two terms, as at and h,
are given, a third arithmetical proportional to them may
be easily found, in order to form an arithmetical pro-
gression. For if we suppose the required term to be
Xy we shall have :
' .' a • b • X
C a • b '.' b • X
Then <a + x=64-^ = 26 (20)
lxzz2b^a: (24)
Consequently, to find a third arithmetical proportional
to two given terms, we must subtract the first from
double the second. Thus, the third arithmetical pro-
portional to 3 and 7, will be 14 — 3= 11; and indeed
V 3 • 7 . 11.
An arithmetical mean proportional between two
given terms, such as a ai;Ld &, may be found with equal
ease ; for if the required mean be denoted by :», we
shall have
• •
a . « > i (20)
(a + bz= 2x
1-2-=* ^^^
Which indicates, that afi arithmetical mean proportional
to two quantities, is equal to the half of these quanti-
ties. Thus, the mean proportional between 9 and 13,
Ull; forv 9 . 11 . 13.
16 iMbsklMtEKTS.
THEOREM II.
24. In every eveii aritnmetical progression, tl>e sura
bt ail the terms, ec^lially distant from the extremes,
taken two and two, is equal to that of the extremes ;
and if it be odd, the sum of the extremes, or of any
two terms equally distant from the extremes, is the
double of the mean term. ^
CAI^E t.
If ^3 -5. 7 . 9 . 11 . 13,
^j^ C5 + ll=3 + 13
inen^^ ^ 9=3+13
CASE II.
If -^2. 4. ,6 . 8 . 10,
|4+8=;2x6
In the first case, the sum of an arithmetical progrcs-
$ion, is equal to the product of the sum of the ex-
ti'eitnes miiltiplied* by half the number of terms ; and in
the second, to the product of the mean multiplied by
the number of terms.
TBEOREM III.
23. In every arithmetical progression, any term
whatever is equal to the first and as many times th«
common difference as there are terms before it.
I f V 2-5.8 . 11 . 1 4,
{14 == 2 + 3x4
11-2 + 3x3
8=2^3x2
It hence follows, that we may easily find the vald^
of any term of an arithmetical progression, the first
term, the common difference, and the number of terms
of which are known.
* *li
INTRODUCTION. 17
For example, the 121st term of an arithmetical pro*
^ession^ the first term of which is 5 and the common
dMTerence 3, will be 365 ; for 5 + 3 x 120 zz 365.
Properties of Geometrical Proportion and Progremon,
•► THEOREM I.
^4. In every geometrical proportion, the product of
the extremes is equal to that of the means. .
If 3:6 :: 4 : 8
Then3 x 8 z= 6 x 4
Consequently,' the fourth term of a geometrical pro-
portion, the other vthree of which are known, may be
easily found ; for if the required term be an extreme,
it will be equal to the product of the means divided
by the other extreme ; and if it be a mean, it will be
equal to the product of the extremes divided by the
other mean.
If 2 : 4 :: 3 : r
r2x=4 X 3 (15)
Then< 4x3
= 6
I 2
If 2 15/ :: 3: 6
(2x6 = 33^ (15) ^ ,
Then<2 x 6
|— 3~=5/ = 4
Two terms being given, a third, geometrically propor-
tional to them, may be .easily found, in order to form a
geometrical progression. Let us suppose that a third
proportional is required to the terms a and ft, and
that the term sought is denoted by y. We shall then
have
Then
18 AMUSEMENTS.
Consequently, to find a third term, geometrically pro-
portional, we must divide the square of the second, or
Its product by itself, by the first term. Thus the third
geometrical proportional to 3 and 6, will be — - — •
= 12; and indeed -H- 3 : 6 : 12.
A mean geometrical proportional between two terms,
as a and by may be found with equal ease ; for if this
term be called x, we shall have:
J-
'^ a I X : b
XX or x^ zz ab
Then -{ >^ xx=i ^/ ab
xzz h/ ab
Thus, if we suppose a zz 2, and b zz S; x will be
equal to the square root of 16, which is 4. And indeed
TT 2 : 4 : 8.
THEOREM II.
25. Any term whatever of a geometrical progression,
is equal to the product of the first term multiplied by
the common ratio, raised to that power the exponent
of which is equal to the number of terms before it.
Let the geometrical progression be -H- 2 : 4 : 8 : 16
: 32 : &c.
The fifth term 32, for example, is equal to the pro-
duct of 2, the first, multiplied by 16, which is the fourth
power of the ratio 2.
THEOREM III.
26. In every geometrical progression, the second
term, less the first, is to the first, as the last, less the
first, is to the sum of all those which precede it.
If TT 2 : 4 : 8 : 16 : 32 : &c.
Then4 — 2 : 2::32 — 2 : 2 + 4 + 8 + 16.
INTRODUCTION. 19
RULE OF THREE.
27. The Rule of Three, is an operation by which,
when three terms of a geometrical proportion are
known, a fourth, not known, may be found ; and it is
called direct when the similar terms increase in the
same ratio. For example : if four men perform six
yards of work in a certain time, it is evident that a
greater number must perform more in the same time.
On the other hand, if the similar terms, instead of in-
creasing in the same ratio, must decrease, the rule is
C2Jied inverse, as is the case in the following example:
If four men perform a certain work in eight days, a
greater number of men must perform it in a time pro-
portionally less.
The Rule of Three Direct, and the Rule of Three
Inverse, may be expressed by the following formulee.
Men Men Yards Yards
t) '*
X
Men Men Days Days
3 : 6 :: X : 8
Tlie Rule of Three is compound or simple, according
as the terms are compound or simple. For example,
the above two formulee express each the simple ruk of
three. But if it were required to divide the profits of
a commercial company among several partners, who
have advanced certain capitals, for different periods of
time, it would be necessary to multiply the capital of
each partner by the time, which would render the rule
the compound rule of three.
As the rule of three is only the application of the
formulee of Theorem I. (23.) it is needless to enlarge
farther on this subject.
•23 A'MtJSEMENTS
ARITHMETIC.
ARITHMETIC and Geometry, according to Plato,
are the two wings of the mathematician ; and, indeed,
the object of all mathematical questions, is to deter-
.mine the ratios of numbers or of magnitudes. It may
even be said, to continue the comparison of the an-
cient phtlosopher, that arithmetic is the mathemati*
cian's right wing; for, it is certain, that geometrical
-determinations would often afford very little satisfac-
tion to the mind, if the ratios thus determined, conld
not be reduced to numerical ratios. This justifies the
common practice of beginning with arithmetic.
This science presents a wide field for speculation
and curious research; but in the present selection, wc
•hall confine ourselves to such things as are best cal-
culated to excite the curiosity of those who have a
taste for the mathematics, and who seek for recrea-
tions that may enable them to resume their more serious
studies with greater success .
OF OUR NUMERICAL SYSTEM, AND THE
DIFFERENT KINDS OF ARITHMETIC.
It has been generally observed, that all the nations
with which we are acquainted, reckon by periods of
tea; that is to say, after having counted the units, as
far as ten, they begin again by adding units to a ten ;
when they attain to 20, they add units as far as -30, or
three tens, and so on in succession, till they come to
100, or ten tens; often times a hundred they form a
thousand, &c. Did this arise from necessity? was it
occasioned by any physical cause ? or was it merely
the' effect of chance ?
No one, after the least reflection, will be inclined to
ascribe it to chance. It is not only probable, but
IN ARITHMETIC. 21
might also be proved, that this system derives its origin
from our physical conformation AH men have ten
fingers, a very few excepted, who by some lusus natunt
have twelve. The first men began to reckon on their
fingers. After having exhausted them by counting the
units,^ it was necessary for them to begin and to count
them again, till they had exhausted theoa a second
time ; then a third time, and so on. Hence the origin
of ten; which, being confined to the fingers, could not
be carried beyond the number of ten without forming
a new total, called a hundred; then another, consist-
ing of ten hundreds, called a thousand, <S^c.
A curious consequence hence follows; which is, that
if nature, instead of ten fingers, had given us twelve,
our numerical system would have been different. Af-
ter ten, instead of saying ten plus one, or eleven, w^
•bould have ascended by simple denominations to
twelve, and should have then counted twelve plus one,
twelve plus two, &c. as far as two dozen. Our hun-
dred would have been twelve dozens, a thousand^ times
twelve dozens, &c.
A six-fingered people, in all probability, would have
had an arithmetic of this kind, which indeed would
not have been inferior to that now in use, or rather
would have been attended with some advantages, which
our present numerical system does not possess.
This method of numeration would have been as ex-
peditious, and even more so than that now universally
received. The number of characters, which would hav«
been encreased only by two, to express ten and eleven,
would have been as little burthensome to the meniory,
as the present characters; and this system would have
possessed some advantages, which ought to give us
reason to regret that it was not originally adopted.
This, however, would no doubt have been the case,
had philosophy presided when the system was first
formed; as it would have readily been seen that 12, of
all the numbers almost between 1 and 20, possesses
the advantage of being small, and of having the great-
est number of divisions ; viz. 2, 3, 4, and 6, by which
it can be divided without a remainder. Besides^ in this
' r'
32 AMUSEMENTS
system the periods of numeration would have had the
advantage of being divisible, the first from one to
twelve, by 2, 3, 4, 6; the second, from one to a hun-
dred and forty -four, by 2, 3, 4, 6, 8, 9, 12, 16, 24, 36,
48, 72; whereas, in our system, the first period from
one to ten has only two divisors, 2 and 5 ; the second
from one to a hundred has only seven, viz. ,2, 4, 5, 10,
20, 25, 50, consequently fractions would have less
frequently occurred in numerical operations.
But what would have been particularly convenient in
in this mode of numeration, is, that it would have in-
troduced into all measures the duodecimal divisions
and subdivisions Thus, as the foot is divided into
12 inches, the inch into 12 lines, and the line into
twelve points; the pound, in the like manner, would
have been divided into 12 ounces, the ounce into 12
drams, the dram into twelve grains, or other denomi-
nations at pleasure ; the day would have \^een divided
into 12 portions, called hours ; the hour into 12 others,
which would have been equal to 10 minutes ; and each
of these into 12 inferior parts ; with so on in succes-
sion.
Should it be asked, what would have been the ad-
vantage of this division, we might reply as follows:—
It is well known, that when it is necessary to divide any
measure into 3, 4, or 6 parts, a whole number of mea-
sures of the lower denomination cannot always be
found, or are found only by chance. Thus, the third,
or tenth of a pound avoirdupois does not always give
an exact number of ounces ; and the third of a pound
sterling does not give an exact number of shillings.
The case is the same with the bushel, and the greater
part of other measures. These inconveniences, which
render calculations complex, would not have occurred,
had the duodecimal progression been universally
adopted. *
Stevin, a Dutch mathematician, proposed to adapt
the divisions and subdivisions of all measures to the
system of numeration sirice adopted by the French,
making them to decrease in decimal progression. Thus,
the fathom would have contained 10 feet, the foot 10
IN ARITHMETIC. 23
inches, the inch 10 lines, &c. This method, however,
though attended with some advantages, is less perfect
tkan the duodecimal, as it gives rise to a greater number
of fractions
I A great many systems of arithmetic have been pro-
posed, such as the binary, ternary, quarternary, &c; •
and even the duodenary; but there is no great reason
to believe that any of them will ever be admitted into
practice.
But we shall add nothing farther on the subject ; for,
as useful recreations are the object of this collection,
we must exclude from it every thing too complex, or of
too frivolous a nature.
We suppose that the reader is sufficiently well ac-
quainted with arithmetic, both in whole and fractional
numbers (2.) to be able to comprehend every thing that
relates to it.
OF SOME PROPERTIES OF NUMBERS.
Under this head, we do not comprehend those pro-
perties of numbers which engaged so much the atten-
tion of the ancients, and to which they ascribed so many
Mysterious virtues. Every one, whose mind is not
\inctured with credulity, must laugh to think of the
good canon of Cezene, father Bungo, collecting into a
f^uarto volume, entitled De Mysteriis Numerorum, all
the ridiculous conceits which Nichomachus, Ptolemy,
Porphyrius, and several more of the ancients, childishly
published in regard to numbers. How could it enter
the minds of reasonable beings to ascribe physical
energy to things purely metaphysical ? Far numbers
are mere conceptions of the mind, and consequently
can have no influence in nature.
None, therefore, but old women, or persons of weak
minds, can believe in the virtues of numbers. Some
entertain a notion, that if 13 persons sit at the same
table, one of them will die in the course of the year;
but there is more probability that one will die in the
same time, if the number be 24,
The series 12345679 is of such a nature,
that it may be multiplied by certain numbers, the pro-
IM? AMUSEMENTS
duct of which shall consist of twos, threes, or fours, &c.
at pleasure.
To find a multiplier which shall give similar figures
in the product, 9 must be privately multiplied by 2, 3,
or 4, according as it is required to have twos, threes.
Of fours, in the product.
For example, if it be required that the product
tl^all contain only twos, we must multiply 9 privately
by 2, which will give 18 for the multiplier of the series.
If we multiply the same number by 3, we shall have 27;
if by 4, ,we shall have 36, &c.
If the series, therefore, be multiplied successively
by 18, 27, 36, the products will be composed of twos,
threes, fours, &c. as may be easily proved by trial. X
If the number 37 be multiplied by any of the terms
©f the arithmetical progression 3, 6, 9, 12, &c. all
the products will consist of similar figures.
37 37 37
3 6 9
X
ill 222 333
We may here observe, that the product 111, is com-
posed of 37 multiplied by 3 ; or of 7 multiplied by 3
wliich makes 21, and 30 multiplied by 3 which maJces
90 ; the sum total being one and eleven tens, which,
according to the laws of numbers, can be expressed,
only by three units.
It will not then appear astonishing, that 37 multi-
plied by 6, should give the double of these three units;
and so of the rest.
The number 5 has this peculiar property, that wheu
nultiplied by an odd number, the product will .always
terminate wi*:h 5 ; and if multiplied by an even number,
• will terminate with a cipher,
5 5 5 5
3 5 4 6
15 25 20 30
The number 9 has this property, that the sum of
IN ARITHMETIC. 25
the figures of every number, of which it is a multiple,
forms a multiple of 9.
Thus, the product of 17 multiplied by 9 is 153, and
the sum of these figures 1 + 5 + 3 zi 9. In the like
manner, the sum of the figures 6777, which is the pro-
duct of 753 multiplied by 9, is equal to 27, a multiple
of9.
This will not appear astonishing when we reflect,
that twice 9 is equal to 18, three times 9 zz. 27, &c.;
where it is evident that the tens and units of the pro-
duct are always reciprocal complements of 9.
If we take any two numbers whatever, one of them,
or their sum, or their difference, will always be divisi-
ble by three. This may be so easily proved, that it is
needless to illustrate it by examples.
Every square number must necessarily terminate
with two ciphers, or by 1, 4, 5, 6, 9; and this may
enable us to determine, at one view, whether a nume-
rical quantity be a square or not. (10.)
The number 2, of all the whole numbers, is the only
one the sum and product of which are equal. Thus 2
4- 2 =; 4, as well as 2 x 2. But in fractional num-
bers we find other two quantities, the sum and pro-
duct of which are in like manner equal.
For this purpose, the sum of the two numbers must
be divided by each of them separately. The two frac-
tions which thence arise, will give the same quantity,
both when added and multiplied. (5. 7.)
If we take the two numbers 2 and 5, and divide
their sum by each of them separately, the two frac-
tions ^, ^y will give the same result when added, as
well as when multiplied. This may be easily proved
' by any person in the least acquainted with vulgar frac-
tions. (2.)
Every square number (10.) is divisible by 3, or be-
comes so when diminished by unity. This may be
easily proved with any square number whatever. Thus,
4—1, 16— 1, 25— 1, 49 — 1, 121- 1, &c. are all
divisible by 3 ; and the case is the same with the rest.
Every square number is divisible by 4, or becomes.
fo when diminished by unity.
c
36 AMUSEMENTS
Every square number is divisible also by 5, or be-
comes so when increased or diminished by unity.
Every odd square, diminished by unity, is a mnU
tiple of 8.
Every power of 5 terminates with 5, and every power
of 6 with 6. (10.)
PROBLEM.
To^d Two Numbers, the Squares of which, if added to^
gether, shall form a Square Number.
If any two numbers whatever be multiplied together,
the double of their product will be one of the two num-
bers sought ; and the difference of their squares will be
the other.
Thus, if the numbers 2 and 3, the squares of which
are 4 and 9, be multiplied together, their product will
be 6; if we then take 12, the double of this product,
and 5 the difference of their squares, we shall have two
numbers, the sum of the squares of which will be a
square number; for their squares are 144 and 25,
which by addition give 169, the square of 13.
OF ARITHMETICAL AND GEOxMETRICAL PRO-
GRESSION, WITH SOME PROBLEMS WHICH
DEPEND ON THEM.
SECTION I.
Arithmetical Progression, with an Explanation of its
Principal Properties,
Any series of numbers, continually increasing or de-
creasing by the same quantity, forms what is called an
arithmetical progression. (19.)
Thus, the series of numbers 1, 2, 3, 4, 5, 6, &c. or
1,5, 9, 13, &c. or 20, 18, 16, 14, 12, &c. or 15,
12, 9, 6, 3, are arithmetical progressions; for, in the
first, Uie difference between each term and the follow*
n% one, which exceeds it, is always 1 ; in the second
;
IN ARITHMETIC. ^
it 18 4; it is 2 also in the third, which goes on decreas-
ing, and 3 in the fourth.
From this definition of arithmetical progression, the
following consequences may be deduced :
1st. Any term of an arithmetical progression, is
equal to the first, plus the common difference taken as -
many times as there are term^ before it. (22.)
2nd. The sum of the extremes is always equal to
the sum of any two terms equally distant from them ;
or double the mean term, if the progression contains
an odd number of terms. (21.)
3rd. If the sum of the extremes be multiplied by
half the number of terms, when the terms are even, or
the mean by the whole number of terms when the lat-
ter are odd, the product will be the sum of the progres-
sion.
By considering, with a little attention, the following
progressions, the truth of these consequences will be
readily perceived.
-T- 2 . 5 . 8 . 11 . 14 . 17.
•T- 1 • 3 . 5 . 7.9.
PROBLEM I.
If a hundred stones are placed in a straight line, at the
distance of a yard from each other, the Jirst being at
the same distance from a basket ; how many yards must
the person walk, who engages to pick them up, one by
one, and to' put them into the basket?
It is evident that, to pick up the first stone, and put
it into the basket, the person must walk two yards;
for the second he must walk 4; for the third 6; and
so on increasing by two, to the hundredth.
The number of yards, therefore, which the person
must walk, will be equal to the sum of the progression
2,4, 6, &c. the last term of which is 200. (22.) But
the sum of the progression is equal to 202, the sum
of the two extremes, multiplied by 50^ or half the
number of terms; that is to say, 10,100 yards, which
makes more than 5| miles.
c 2
S8 AMUSEMENTS
PROBLEM II.
A gentleman employed a bricklayer to tink a wellf and
agreed to give him at the rate of three shillings for
the first yard in depth, 5 for the second, 7 for the
third, and so on, increasing to the twentieth, where he
expected to find water: how much was due to the
bricklayer when he had completed the work 1
This question may be easily answered by the rules
before given, for the difference of the terms is 2, and
the number of terms 20; consequently, to find the
twentieth term, we must multiply 2 by 19, and add 38,
the product, to the first term 3, which wiU give for the
twentieth term 41. (22.)
If we then add the first and last terms, that is to
say, 3 and 41, which will make 44, and multiply this
sum by 10, or half the number of terms, the product
440 will be the sum of all the terms of the progres-
sion, or the number of shillings due to the bricklayer,
when he completed the work. (21.)
He would therefore have to receive 221.
PROBLEM III.
A merchant being considerably in debt, one of his credit
tors, to whom he owed ;61860, offered to give him an
acquittance, on condition of his agreeing to pay the
whole sum in twelve monthly installments ; that is to
say, ^100. the first month, and to increase the pay-
mentby a certain sum each succeeding month to the
twelfth inclusive, when the whole debt would be dis-
charged. By what sum was the payment of each month
increased?
In this problem we have given the first term 100,
the number of the terms 12, and their sum 1860; but
the common difference of the terms is unknown.
This difference may be found in the following man-
ner : — As the sura of the extremes, in an even arith-
metical progression, is equal to the sum total, divided
IK ARITHMETIC. S9
by half the number of tenns, if the sum total 1860 be
divided by 6, or half the number of terms, we shall
have 310 for the sum of the first and last term, from
which if we subtract 100, the first term, the remainder
210 will be the last term; but the last term is always
equal to the first and the common difiference taken as
many times as there are terms before it. If we, there-
fore, deduct the first term 100, from 210 the last, and
divide 110, the remainder, by 11, we shall have 10 as
the required difference. The first term being 100, the
second therefore will be 110, the third 120, &c. (21.)
PROBLEM IV.
A gentleman employed a bricklayer to sink a well, to the
depth of 20 yards, and agreed to give him £20. for
the whole; but the bricklayer happening to die when
he had completed only 8 yards, how much was due to
his heirs ?
To imagine that two fifths of the whole sum were
due to the workman, because 8 yards are two fifths of
the whole depth, would be erroneous; for as the dif-
ficulty must increase arithmetically as the depth, it is
natural to suppose that the price should increase in the
same ratio.
To resolve this problem, therefore, £20, or 400
shillings, must be divided into twenty terms in arithme-
tical progression; and the sum of the first eight of these
will express what was due to the bricklayer for his '
labour.
But 400 shillings may be divided into twenty terms
in arithmetical progression a great many different ways,
according to the value of the first term, which is here
undetermined : if we suppose it, for example, to be 1
shilling, the progression will be 1, 3, 5, 7, &c. the last
term of which will be 39; and consequently the sum of
the first eight terms will be 64 shillings.
But to resolve the problem in a proper manner, so as
to give to the bricklayer his just due fox xVv^ cwa.-
« 3
30 ▲MUS£M£NTS
mencement of the work, we must determifie what is the
£Edr value of a yard of work smiilar to the firsts and
then assume that value as the €rst ter^l of the pro-
gcessioQ. We shall here suppose that this value is 5
shillings ; and in that case the required progression
will be 6, 611., g 3 g^^ n^^ 124!^, &c. the com-
mon difference' of which is l-i4, and the last term
is 35.
Now to find the eighth term, which is necessary
before we can find the sum of the iirst eight terms,
multiply the common difference -f|. by 7, which will
give ll-rV, and add this product to 5, the first term,
which will give the eighth term 16^^; if we then add
16^ to the first term, and multiply the sum, 21^^, by
4, the product, 84i^, will be the sum of the first eight
terms, or what was due to the bricklayer for the part
M' the work he had completed. The bricklayer, there-
fore^ had to receive 84^^ shillings, or £4. 4s. 24fd»
SECTION II.
Of Geometrical Progressions^ with au Explanation of^
their principal Properties,
If there be a series of numbers, each of which »
the product of the preceding by a common multiplier
(18.) these numbers form what is called a geometrical
progression. Thus, 1 2, 4, 8, 16, &c. form a geo-
Bietrical progression ; for the second is the double of
die first, the third the double of the second, and so
on m succession. The terms 1, 3, 9, 27, 81, &c.
form also a geometrical progression, each being th^
triple of the preceding.
Progressions may be either increasing, as the two
above mentioned, or decreasing as the two following
16, 8, 4, 2, 1 ; 81, 27, 9, 3, 1.
The principal property of geometrical progression is^
that if we take any three following terms whatever^
as 3, 9, 27, the product 81 of the extremes will be
equal to the square of the mean 9. In like manner,
if ve take any four following terms^ as 3, 9, 27^ 91 ^
IN ARITHMETIC. 31
th6 product 243, of the extremes, will be equal to
that of the two means, 9 and 27.
In the last place, if any number of terms whatever
of the series be assumed, as 2, 4, 8, 16, 32, 64, the
product of the extremes, 2 and 64, will be equal to
the product of any two terms equally distant from
them, as 4 and 32, or 8 and 16. If the number of
terms be odd, it is evident that there will be only one
term equally distant from the two extremes ; and in
that case the square of that term will be equal to th^
product of the extremes, or of any two terms whatever
equally distant from them, or from the mean. (23. 24.)
Between geometrical and arithmetical progression,
there is a certain analogy, which deserves here to. be
mentioned, and which is, that the same results are
obtained in the former, by employing multiplication
and division, as are obtained in the latter by addition
and subtraction. When, in the latter, we take the
half or the third, we employ in the former extraction of
the square, cube, &c. roots.
Thus, to find an arithmetical mean between two num-
bers, for example 3 and 12, we must add the two ex-
tremes together, and take the half of 15 their sum,
which is 7| ; but to Bnd a geometrical mean between
two numbers, we must multiply them together, and ex-
tract the square root of their product ; for example, if
a geometrical mean between 3 and 12 be required, we
must extract the square root of their product 36, which
will give 6; and, indeed, rr 3 : 6 : 12. (23.)
A geometrical progression may decrease in infinitum^
without ever coming to ; for it is evident that any
part of a quantity whatever, greater than can never
become 0. A decreasing geometrical progression may
be continued, therefore, in infinitum^ since to find the
following term, nothing will be necessary but to divide
the last term by the exponent or common ratio. We
shall here give two examples of decreasing geometrical
progressions.
ii, i.i.i. t • I • 1 jiyf,
• • * • T • y • "57 • IT • 7TT> *^^'
c 4
3d AfiltJSEMENTd
The sum of an increasing geometrical progre^siotfy
continued ad infinitum, is evidently infinite ; but that of
a decreasing geometrical progression, whatever be the
number of terms supposed, is always finite. Thus,
the sum of the terms, continued in injlnitumy of the
geometrical progression ^ 1 : J : J ; 4~t5 ! &c. is
only 2. That of the progression ttI : t • i • tt • tt •
&c. is only Ij.
OF HARMONICAL PROGRESSION.
Three numbers are in harmonical proportion, when
the first is to the last as the difference between th«
first and the second is to the difference between the
second and third. Thus the numbers 6, 3, 2, are in
harmonical proportion; for 6 is to 2 as 3, the difference
between the two first numbers, is to 1 , the difference
of the two last. This kind of relation is called harmo-
nical, for a reason which will be seen hereafter.
When three numbers in decreasing harmonical pro-
portion are given, it is easy to find a fourth ; nothing is
necessary but to find a third harmonical to the two last,
and this will be the fourth term required. In like man-
ner, the third and fourth may be employed to find a
fifth, and so on in succession: this will form what is
called a harmonical progression, which by the above
method may be always continued decreasing.
If we suppose the two first numbers to be 2 and I,
we shall have the harmonical progression 2, 1, J,"'', J,
|., ^, &c. It is a remarkable property, therefore, of
the series of fractions, having unity for their numera-
tors, and for their denominator^ the numbers of the
•natural progression, that they are in harmonical pro-
gression.
This series of numbers, indeed, contains all the mu-
sical concords possible; for the ratio of 1 to J, gives
the octave ; that of } to i, or of 3 to 2, the fifth ; that
of 4- to J, or of 4 to 3, tne fourth; that of J to -J, or
of 6 to 4, the third major ; that of \ to ■^, or of 6
' to 5, the third minor ; that of ^ to ^, or of 9 to
8, the tone major; and that of ^ to i^, or of 10 to 9>
IK ARITHMETIC. 33
the tone minor. But this mVL be explained more at
large when we come to treat of harmony.
Let us now return to geometrical progression^ and
the application of it to a few problems, which may
serve as a rule for the solution of all others of the
same kind.
PROBLEM I.
If Achilles can walk ten times as fast as a tortoise , which
is a furlong before him, can crawl, will the former over-'
take the latter ; and how far must he walk before he
does so? .
This question has been thought worthy of notice
merely because Zeno, the founder of the sect of the
stoics, pretended to prove by a sophism, that Achilles
would never overtake the tortoise ; for while Achilles,
said he, is walking a furlong, the tortoise will have
advanced the tenth of a furlong ; and while the former
is walking that tenth the tortoise will have advanced
the hundredth part of a furlong, and so on in infinitum;
consequently, an infinite number of instants must elapse
before the hero can come up with the tortoise, and
therefore he will never come up with it
Any person, however, of common sense may readily
perceive, that Achilles will soon come up with the tor-
toise. In what then consists the sophism? It may be
explained as follows:
Achilles, indeed, would never overtake the tortoise,
if the intervals of time, during which he is supposed
to be walking the first furlong, and then the tenth,
hundredth, and thousandth parts of a furlong, which
the tortoise has successively advanced before him, were
equal : but if we suppose that he has walked the first
furlong in 10 minutes, he will require only one minute
to walk the tenth of a furlong, and T^y of a minute to
walk the hundredth, &c. The intervals of time, there-
fore, which Achilles will require to pass over the space
gained by the tortoise during the preceding time, will
c 5
S4 AMUSEMKNTS
go on decreasing in the following manner : 10, 1 , -^ y^
7^^, &c. ; and Uiis series fonns a sub-deca|^ geome-
trical progression; the snm of which is equal to 11-^, or
the interval of time^ at the end .of which Achilles will
have reached the tortoise. -
PROBLEM II.
If the hour and minute hands of a clock both begin to mote
exactly at noon, at what points of' the dial-plate uUl
they be successively in conjunction, during a wko& reco-
lutioH if the twelve hours ?
This problem, considered in a certain manner, is in
nothing different from the preceding. The minute
hand acts here the part which Achilles did in the for*
mer; and the hour hand, which moves ten times slower,
that of the tortoise. In the last place, if we suppose
the hour hand to be beginning a second revolution, and
the minute hand to be beginning a first, the distance
which the one has gained over the other will be a
whole revolution of the dial-plate. When the minute
band has made one revolution, the hour hand will have
made one twelfth of a revolution, and so on progress
sively. To resolve the problem, therefore, we need
only apply to these data the method employed in the
former case, and we shall find, that the interval from
noon to the point where the hands come again into
conjunction will be iV of a whole revolution; or, what
amounts to the same thing, one hour and |V of an hour.
They will afterwards be in conjunction at 2 hours and
-X ; 3 hours and I'r » ^ hours and Vt> &c* and, in the
last place, at 11 hours and H, that is to say, at 12
hours.
PROBLEM III.
A sovereign f being desirous to confer a liberal reward on
one of hi* courtiers, who had performed some very m-
porttmt service, desired him to ask whatever he thought
proper, assuring him it should be granted. The courtier.
IK AIUTHMETIC* 35
who roas well aequmnted 'with the science of mtmhersj
mdy requested that the monarch would give mm a ^tion-
tiiy of wheat equal to that which woM arise from one
grain doubled sixty^three times successivdy. What was
the value of the reward?
It will be found by calculation that the sixty-fourth
term of the double progression rf 1 : 2 I 4 : 8 : 16 :
32 : &c. is 922337^036854775808. But the sum of
all the terms of a double progression, beginning with
unity, may be obtained by doubling the last term, aad
subtracting from it unity. The number of the grsdns of
wheat, therefore, in the present case will be 184467440
73709551615. Now, if a standard pint contains 9216
grains of- wheat, a gallon will contain 73728 : and, as
eight gallons make one bushel, if we divide the abo?e
result by eight times 73728, we shall have 31274997
.411295, for the number of the bushels of whe^t equal
^to the above number of grains, a quantity greater than
what the whole surface of the earth could produce in.
several years, and which, in value, exceeds all the richei^
perhaps on the globe of the earth.
Another problem of the same kind is proposed in the
following manner: .
A gentleman, talcing a fancy to a horse^which a horse-dealer
wished to dispose of at as high a price as he could^ the
latter, to induce the gentleman to become a purchaser,
offered to let him have the horse for the value of the
twenty-fourth nail in his shoes, reckoning one farthing
for the first nail, two for the second, four for the third,
and so on to the twentyfourth. The gentleman, thinking
he should have a good bargain, accepted the offer : what
was the price of the horse ?
By calculating as before, the twenty-fourth term of
the progression -rr 1 : 2 : 4 : 8 : &c. will be found to
be 8^888608, equal to the number of farthings the pur-
diater gave for the horse. The price therefore,
amoantMl to £8738. 2s. 8d. which is more than asy
Arabian horse, even of the noblest breed, was tfts
•oU for. .
c6
36 AMU6CM£KT«
We shall conclude this article with some phyifeica*
mathematical observations on the prodigious fecutidity
and progressive multiplication of animals and vegeta-
bles, virhich would take place if the powers of nature
were not continually meeting with obstacles.
Isc. It is not astonishing, that the race of Abraham,
' after sojourning 260 years in Egypt, should have formed
a nation capable of giving uneasiness to the sovereigns
of that country. We are told, in the sacred writings,
that Jacob settled in Egypt with seventy persons: now,
if we suppose, that among these seventy persons, there
were twenty too far advanced in hfe, or too young to
have children; that of the remaining fifty, twenty-five
were males and as many females, forming twenty-five
married couples, and that each couple in the space of
twenty-five years, produced, one with another, eight
children, which will not appear incredible in a country
celebrated for the fertility of its inhabitants ; we shaU
find that, at the end of twenty- five years, the above
seventy persons may have increased to two hundred
and seventy ; from which. If we deduct those who died,
there will, perhaps, be no exaggeration in making them
amount to two hundred and ten. The race of Jacob,
therefore, after sojourning twenty-five years in Egypt,
may have been tripled. In like manner, these two
hundred and ten persons, after twenty-five years more,
may have increased to six hundred and thirty ; and so
on in triple geometrical progression: hence it follows,
that, at the end of two hundred and twenty-five years,
the population may have amounted to 13778 10 persons,
among whom there might easily be five or six hundred
thousand adults fit to bear arms. " "^"
2nd. If we suppose that the race of Adam, making
a proper deduction for those who died, may have been
doubled every twenty years, which certainly is not in-
consistent with the powers of nature, the number of
men at the end of five centuries, may have amounted
to 1048576. Now, as Adam lived about 900 years,
he may have seen, therefore, when in the prime of life,
a posterity of 1048576 persons.
3rd. How great would be the multiplication. of many
IN ARITHM£tIC. B7
ftnimals, did not the difficulty of finding food, the con*
tinual war which they carry on against each other, Or
the numbers of them consumed by man, set bounds for
their propagation ! It might easily be proved, that the
breed of a sow, which brings forth six young, two males
and four females, if we suppose that each female pro-
duces every year after six young, four of them females
and two males, would in twelve years amount to
33554230.
Several other animals, such as rabbits and cats, which
go with young only for a few weeks, would multiply
with still greater rapidity; in half a century the whole
earth would not be sufficient to supply them with food,
nor even to contain them.
If all the ova of a herring were fecundated, a very
few years would be sufficient to make its posterity fill-
the whole ocean; for every oviparous fish contains
thousands of ova, which it deposits In spawning time.
Let us suppose, that the number of ova amounts only
to 2000, and that these produce as many fish, half males
and half females ; in the second year there would be
more than 200000 ; in the third, more than 200000000;
and in the eighth year, the number would exceed that
expressed by 2 followed by twenty-four cyphers. As
the earth contains scarcely so many cubic inches, the
ocean, if it covered the Whole globe, would not be suffi-
cient to contain all these fish, the produce of one herring
in eight years !
4. Many vegetable productions, if all their seeds
were put into the earth, would, ui a few years cover the
whole surface of the globe The hyosciamus, which,
of all the known plants produces, perhaps, the greatest
number of seeds, would, for this purpose require no
more than four years. According to some experiments,
it has been found that one stem of the hyosciamus
produces sometimes more than 50000 seeds : now, if
we admit the number to be only 10000, at the fourth
crop it would amount to a 1 followed by sixteen cy-
. phers. But, as the whole surface of the earth contains
no more than 5507634452576256 square feet, if we al-
low to each plant fcly one* square foot, it will be seen
38 AMUSEMENTS
that the whole surface of the earth would not be suffix
cient for the plants produced from one hyosciamus at
the end of the fourth year I
EXERCISES IN THE SINGLE AND COMPOVND
RULE OF THREE, BOTH DIRECT AND IN-
VERSE.
(26.) We shall confine ourselves to a small number
of examples in this rule, which we briefly explained
in the introduction.
SINGLE RULE OF THREE DIRECT.
EXAMPLE I.
IfAO pioneers can dig a trench 268 yards long, in a cer*
tain time^ how many yards can 60 pioneers dig in the
same time ?
40 ; 60 : : 268 : X = 402. (23.)
EXAMPLE II.
If a ship with a fresh breeze, sails 200 leagues in three
days^ how long time will she require ^o^at/2000 leagues^
every other circumstance being the same ?
200: 2000:: 3 :j: = 30.
EXAMPLE III.
If 52 yards 2 feet and 5 inches of mason xoork cost £16S^
9s, id. what will be the expence of 11 yards, 2 feet 8
inches ^ the like kind of work, ai the same rate ?
To render the solution of this ^!^blem easier, the
<|ttantity of each piece of work must be reduced to
inches, by multiplymgthe yards by 3 and 12 ; and, Ibr
the same reason, Uie |^ce of the woilc must be reduced
to {ttnce. We shall then ha?e the follcming propor-
tion: —
IN ARITHMETIC. 39
Indies, Inches. Pence.
1901 : 2804 :: 40432 : x. (23.)
SINGLE RULE OF THREE INVERSE.'
EXAMPLE I.
If 50 men can perform a certain piece of work in 25 days^
kow many men viill he requisite to perform the same work
in 10 dai/s?
It is here evident that, as the 'work is to be done in a
shorter time, it will require more men. Consequently,
the proportion must be expressed in this manner :
Dayi> Days. Men.
10 : 25 ;: 30 : x =z 75.
EXAMPLE II.
A vessel has provisions for 15 dat/s, hut being ohliged by
certain circumstances, to continue at sea for 20 duys, to
what quantity must the daily ration of each man be re-
duced^ to make the provisions last during that time ?
If the quantity of provisions consumed daily be je*
presented by unity, it is evident that the reduced quan-
tity must be as much below 1 as 15 are less than 20.
We shall therefore have
20 : 15 :: i : a? = |.
COMPOUND RULE OF THREE.
EXAMPLE I. ,
If 30 men perform 1 32 yards of work in 1 8 days, how much
will 54 men perform in 28 days f
Thirty men, working 18 days, will perform the same
work as 18 timeft 30 men = 540, in one day; and, in
like manner, 54 men, in 28 days, will perform the same
work as 54 times 28 men z= 1512, in one day. We
lu^ve, therefore, the following proportion : -
540: 132:: 1512 : *.
I.
43
BXAlfPLE It.
If a many walking 7 hours a dea/y travels 230 leagues in.
30 daj/Sy how many days would he require to perform a
Journey of 600 leagues ^ walking 10 hours a day T
This problem may be reduced to the single rule of
' three, if we consider, that to travel 30 days, employing
7 hours each day, is the same thing as to travel 30 times
7 hours, or 210 hours. The question, therefore, may
be changed in the following manner: If 210 hours are
required to travel 230 leagues, how many hours will
be requisite to travel 600 leagues? When the number
of hours which answer the question have been found,
the required number of days may be found by dividing
these hours by 10, as the traveller employs 10 hours
each day. We must, therefore, find the fourth term of
tlie proportion, the first three of which are as follows;
Jsfnti Leiig> Hours. Hours.
230 : 600 :: 210 : x:
RULE OF FELLOWSHIP.
As this rule is merely an application of what has
been said respecting the rule of three, we shall only
give a few examples; to illustrate the use of it.
EXAMPI^E I.
A privateer, belonging to three merchants captured a prize
worth j£80000. what will each partner's share amount to,
the Jirst having advanced to purchase and Jit out the
vessel £2000. the second £6000. and the third £ 1 2000 ?
It is here evident, that each partner must haye a
share of the prize proportioned to the money he ad-
vanced.
Wo must, therefore, make this proportion: As the
•urn advanced by each partner, is to the whole money
advanced, so is the share of each to the whole prize.
Hence we shall have the following proportions, where
m ARlTHMEtlC. 41
the second and fourth terms, that is to say, the sum
total of the money advanced and the value of the prize,
are common to all of them:
6000 i : 20000 : : <x: soooo.
2000
6000
12000 )
EXAMPLE II.
Three persons having entered info partnership, the first
advanced £3000. for six months, the second £4lOQO, far
five months, and the third ^8000. for nine months ; at
the end of that time they found that their gain amounted
to ^150000. how much will each partner^s share he
worth ?
As this problem belongs to the compound rule of
three, we shall take, as the first terms, the product of
the money advanced by each partner, multiplied by the
time it was employed ; and for the second, the sum of
these products, in the following manner :
aoooov : iioooo :: Ix : 150000
18000*)
aoooo \
72000 )
It may be readily seen, that by means of the rule of
three, any sum, such as the amount of a legacy, for
example, may be easily divided among several persons,
in such a manner, that the shares shall be in the ratio
of certain determinate numbers, as 3, 4, 6. In this
case, these numbers must be considered as three sums
advanced by three partners, and their sum as the total
of the money advanced: if we then call the legacy to
be divided a, we shall have the following proportibn :
3
4
6
i : 13 :: <x : a.
4 2 AM USEM CKT8
RULE OF ALLIGATION.
Alligation is of two kiuds. The first consists in find*
ing the common price of several things, supposed to be
mixed together ; as if a goldsmith, for example, should
make a composition of gold, silver, &c. and be desirous
to know the value of an ounce of this mixture. The
same rule is employed to determine the mean price of
several liquors, or different kinds of merchai^dise, mixed
together.
This rule is exceedingly easy; for nothing is neces-
sary, in solving questions of this kind, but to divide
the whole value of the articles by the quantity of each
article eitaployed for the mixture, and the quotient will
be the answer.
EXAMPLE I.
A wine merchant mixes together 200 bottles of Madeira^
at 5 shillings, 500 of Port at 3s. 800 of Malaga at 4*.
and 300 of' Tokay at Bs, ; how much is a bottle of this
mixture worth ?
200 X 5 = 1000
500 X 3 = 1500
800 X 4 = 3200
300 X 8 zz 2400
1800 SlOO shillings.
Now, if 8100 shillings, or the whole value of the wine,
be divided by 1800, the number of the bottles, the
quotient will express the value of each bottle of the
' . r^ , 8100 81 , ^^
mixture. Consequently y— - = yg = 4^. 6d.
EXAMPLE II.
A gentleman employed 300 workmen, 50 of which were paid
at the rate of %s, a day ; 70 at the rate of 6s, and 180
at the rate of As, ; how much did each of them, taking one
with another, cost him per day?
The sum total, which is 1540 shillings, must be di^
vided by 300, the number of the workmen, and the
quotient will be what each of them, taken one with
another, cost him per day.
1540 154 ^ ,
300 30 — ^^
The object of the second kind of alligation, is to de-
termine in what proportion several things, of different
values, ought to be mixed, in order to have an article
of a certain mean price.
To obtain this result, the prices of the things to be
mixed must be arranged, as seen in the following ex-
amples : —
EXAMPLE I.
A grocer, who has tea at 3s. 4«. 7^. and 9s, per pound, is
■ desirous of having a mixture which he can sell at 5s per
pound. In what proportions must he mix these four
kinds of tea; so as to be able to sell the mixture at 5s, ?
Arrange the prices of the things to be mixed as seen
at A; placing those which are greater than the mean
price at the top, and those which are less at the
bottom. \
Fig, A.
3*.
2
4*.
5s,
4
7^.
2
9s.
1
Then compare in succession with the mean prices the
prices of all the things to be mixed, and place the dif-
ferences as in the above example.
Thus, the difference between 3 and 5 is 2, which
must be set down opposite to 7, and that between 4
and 5 is 1, which must be placed opposite to 9. Then
proceed to the prices greater than that of the mean
44 AMUSEMfiNTSI
price, and compare them with that price in the follow-
ing manner : The difference between 7 and 5 is 2, which
place opposite to 3 ; and that between 9 and 5 is 4,
which place opposite to 4.
The right hand column, the sum of ^hich is 9, shews
that to have 9 pounds of tea, at the mean price of 5s,
the mixture must consist of 2 pounds at 35. 4 pounds
at 45. 2 pounds at 7s. and 1 pound at 95.
It may here be readily seen, that 9 pounds of tea, at
the mean price of 5s, will amount exactly to the value
of the quantities mixed.
It may sometimes happen that the figures, expressing
the different values of the things to be mixed, will not
be equal in number both below and above the mean
price; on this account, if there are, for example, three
figures above it, and only two below, the difference of
the third figure at the top must be placed opposite to
the second at the bottom, along with the difference of
the second at the top; and the difference of the second
figure at the bottom must be set down twice ; that is to
say, it must be placed opposite to the second and thfe
third at the top. See fig. B.
Fig,B.
2s.
1
3s,
■^
2
45.
5s,
2
6s,
3
Is.
2+1=3.
11
That is to say, to form 1 1 pounds of tea, of the meart
price of 5s, one pound at 25. two pounds at 35. two
pounds at 45. three pounds at 6s. and three at 75. must
be mixed together.
EXAMPLE II.
A goldsmith has gold of 23 carats fine, and some qf\3 ca^
rats, which he is desirous of mixing, so as to form gold
of IS carats, what quantitu of each must he take ? See
Fig.C.
IN ARITHMETIC. 45
13 5
Fig. C. 18
23 5
10
It hence appears, that he must take an equal quan*
tity of each.
By the same rule, we may find the quantity of alloy
in any compound metal, for example bronze, which con-
sists of copper and tin mixed together in a certain
proportion. For this purpose, we must take three in-
gots of the same weight, one of bronze, another of cop-
per, and a third of pure tin. These three bodies, when
weighed in water, will each lose a different part of their
weight: the ingot of tin will lose more, and that of
copper less, than the ingot of bronze. Let us suppose,
that the loss of the bronze is 3 ounces, that of copper
2^, and that of the tin 3^. If these three numbers be
arranged, according to the above formula, we shall
hare
2J 4
3
3i ^
I
The sum of the two differences, |, shews that, in |
of bronze, there are one of copper and two of tin. This
proportion being found, we may thence conclude that
a mass of bronze similar to the ingot, weighing 150
pounds, would contain 100 pounds of tin, and 50 of
copper.
A CHRONOLOGICAL PROBLEM.
How rnany years^ months^ and days^ elapsed between the
Battle of Marignan, fought on the 3rd of September,
1515, and that of Font enoi, fought on the \lth of May ^
1745?
The period from the Christian era to the 3rd of Sep-
tember, 1515, comprehends 1514 years, 8 months ^.wd
46 AMUSEMENTS
3 days ; and tliat from the same epoch to the 11th of
May, 1745, comprehends 1744 years, 4 months, and 11
days.
Consequently, if we subtract the former from the
latter, the difference, 229 years, 8 months, and 8 days,
will express the interval of time between the battle of
Marig^an and that of Fontenoi.
This method may be employed for every other pro-
blem of the like kind, and especially when it is ne-
cessary, in calculating interest, to know how many
years, months, and days, have elapsed between certain
dates.
THE RULE OF TARE.
\
By tare is commonly meant the weight of the cask,
box, or bag, in which goods are contained, and which
being subtracted, when known, from the gross weight,
leaves the real weight of the goods, called the net
weight. In general, an allowance is made for it, at the
rate of so much per hundred weight ; and the quantity
to be deducted is found by the Rule of Three, as in
the following example :
A merchant pvrckases a bale of cotton weighing 7 cwt» iti"
eluding the package, and is allowed at the rate of 16
per cwt. of tare : how much ought to be deducted on
that account, from the gross weight of the bale of
cotton ?
As the merchant purchases the goods by the net
weight, the seller must give him 16 pounds over and
above each cwt.; that is to say, for each 112 pounds
he must give him 128. We must, therefore, make the
following proportion:
128 : 112 :: 700 : x, (23.)
The fourth term will express the number of pounds
for which the merchant ought to pay.
IN ARITHMETIC. 47
DISCOUNT.
A'merchant purchases goods io the amount of ;f 1000. to
be paid at the end of a year; but the vender offers to
abate 10 per cent, for ready money; how much must
the buyer pay down ?
It might here be supposed, that the abatement ought
to be as many times .£10. as 100 is contained in 1000;
that is to 8ay,.that.£l00. ought to be deducted, so that
the merchant would have to pay only £900.
But it is to be observed, that the vender ought to
allow the purchaser only 10 per cent, on what he will
really receive; that is to say, that every J 10 pounds
which the merchant has to pay, ought to be reduced
to 100. We have, therefore, the following propor-
tion:
110 : 100 :: looo : x.
This is the only true method of estimating discount;
for if the vender received only ^£900. ready money,
this sum at 10 per cent, would produce, at the end of
a year, no more than £990. consequently it would be
much better for him to give a y^ar s credit and receive
£1000.
OF COMBINATIONS AND PERMUTATIONS.
Before we enter on this subject, it will be necessary
to explain the method of constructing a kind of table,
treated of by Pascal and others, calkd the arithmetical
triangle ;' which is of great use to shorten calculations
of this kind.
First form a band A B of ten equal squares, and
below it another C D of the like kind, but shorter by
one square on the left, so that it shall contain only
nine squares; and continue in this manner, always
making each su<5ce8sive band a square shorter. W%
sbaU thus hare a series of squares, disposed in verClc^l
48 AMUSEMENTS
and hoirizontal bands, and termjaatiag at each extre-
mity in a single square, so as to form a triangle, on
which account it has been called the arithmetical tri-
angle.
The numbers with which it is to be filled up, must be
disposed in the following manner:
In each of the squares of the first band inscribe
unity, as well as in each of those iaike diagonal AE,
.l,
1
lj.l
1
1
'
1 1
1
c
'
2
3
4
5
6
.|a
9
1
^
6
10
15
2l|28
36
1
4
10
»
3a\56
84
1
5
15
35
70
126
1
6
21
56
196
1
7
1
28
84
8
I
36
9
1
E
Than add the number in the first square of the band
C D, which is unity, to that in the square immediately
above it, and write down the' sum 2" in the following
square. Add thia number, in like manner, to that in
the square above it, which will give 3, and write it
down in the next square. By these means we shall
have the series of the natural numbers 1,2, 3, 4) 5, &c-
The same method must be followed td fill up the oUier
horizontal bands; that is to say, each aquare ought 01"
in AlRlTHMXmc. 49
ways to contain the sam of the number in the preceding
Square of the same row, and that which is immediately
above it in the preceding. Tlius, the number 15,
which occupies the fifth square of the third band, is
equal to the sum of 10, which stands in the preceding
square, and of 5, which is in the square above it.
The cise is the same with 21, which is the sum of 15
and 6; with 35, in the fourth band, which is the sum of
15and20, &c.
The different series of numbers, contained in this
triangle, have different properties ; but we shall here
speak only of those which relate to combinations and
permutations, as the rest are of too abstract a nature
to be employed in arithmetical recreations, the princi-
pal object of which is to afford easy and agreeable
amusement.
There are two principal kinds of combination. The
first is that where the different arrangements of several
things are sought, without any regard to their change
of place.
The second is that where regard is paid to the differ-
ent changes of place. For example, the three quan-
tities A, B, C, taken two and two, without regard to
the different changes of place, are susceptible of only
three combinations A B, AC, B C; but if we pay re-
gard to the changes of place, they are susceptible of
six combinations; for, besides the three former, we
shall have B A, C A, C B.
In combinations, properly so called, no attention is
paid to the order of the things. If four tickets, for
example, marked A, B, C, D, were put into a hat, and
any one should bet to draw out A and D, either by
taking two at once, or one after the other, it would be
of no importance whether A should be drawn first or
last ; the combinations A D and D A ought, therefore,
to be here considered only as one.
But, if any one should bet to draw out A the first
time, and D the second, the case would be very dif-
ferent, and it would then be necessary to attend to the
order in which these four letters may be taken, and
arranged together, two and two : it may be rea.diV^
50 AUfJ^MfSNTS
seei^, that the different ways are A B, B A, A C, C A,
AD, DA, B C, C 9, B D, D B, C D, D C. Iji
like manner, these fpur letters might be combined and
arranged, thi:ee and dyree, twenty-four ways : as A B C,
ACB, BAG, BCA, CAB, CBA, ADB, ABD,
DBA, DAB, BAD,BDA,ACD, ADC, DAC,
DC A, CAD, CD A, BCD, B D C, C B D,
C D B, D B C, D C B. This is what is called per-
mutation, ot change of order.
PROBLEM I.
Any number of things whatever being given, to determine
in how many d\ff^erent ways they may be combined^ two
' and tzDOf three and three, Sfc, without regard to order.
This problem may be easily resolved by making use
of the arithmetical triangle. Thus, for example, if there
are eight things to be combined, three and three, we
must take the ninth vertical band, or, in all cases, that
band the order of which is expressed by a number ex-
ceeding by unity the number of the things to be comr
bined; then the fourth horizontal band, or that the
order of which is greater by unity than the number of
the things to be taken together, and in the common
square of both will be found the number of the combi-
nations required ; which, in the present example, will
be 66,
But, as an arithmetical triangle may not always be
at hand, or as the number of things to be combined
may be too great to be found in such a table, the fol-
lowing simple method may be followed :
The number of things to be combined, and the man-
ner in which they are to be taken, viz. two and two,
or three and three, being given :
1st. Form two arithmetical progressions, oiie in
which the terms go on decreasing by unity, beginning
with the given number of things to be combined, and
the other consisting of the series of the natural num-
bers 1, 2, 3, 4, &c.
m ARSCHMKBIC. M
2ad. Then take from each as man]f terms as there
are things to be arranged, together, in the proposed
combination.
3rd. Multiply together the terms of the first pro-
gression, and do the same with those of the second.
4th. In the last place, divide the first product by the
second, and the quotient will be the number of the com-
binatioas required;
PROBLEM II.
in how many ways can 90 things be combined^ two and two ?
According to the above rule, we must multmly 90
by 89, Mid divide the product, 8010, by the product of
1 and 2, that is 2 ; the quotient 4005 will be the num-
b^ of combinations resulting from 90 things, taken
imo and two.
Sl^uld it be required to determine, in how many
ways the same things can be combined three and three,
the problem may be answered with equal fease ; for we
have only to multiply together 90, 89, 88, and to di-
ynde the product, 704880, by that of the three numbers
1^ 2, 3; the quotient 117480 will be the number re-
^pured.
hk. like manner, it will be found, that 90 things may
be combined, four and four, 2555190 ways; for if the
product of 90, 89, 88, and 87 be divided by 24, the pro-
duet of 1, 2, 3, 4, we shall have the above result.
Were it asked how many conjunctions the seven
{Janets could form with each other, two and two, we
might reply 21; for, according to the general rule, if
we multiply 7 by 6, which will give 42, and divide that
number by the product X)f 1 and 2, that is 2, the. quo-
tient will be 21.
If we wished to know the number of all the con-
junctions possible of these seven planets, two and two,
three and three, &c. ; by finding separately the number
of the conjunctions two and two, then those three and
three, &c. and adding them together, it will seem that
they amount to 120.
s2
53 AHUBSMEKn
PROBLEM III.
Any number of things being given^ to find in how mang
ways they can be arranged*
This problem may be easily solved by following the
method of induction ; for,
1st. One thing can be arranged only in one way; in
this case, therefore, the number of arrangements is = 1.
2nd. Two things may be arranged together two ways ;
for with the letters A and B we can form the arrange*
ments A B, and B A; the number of arrangemtats,
therefore, is equal to 2, or the product of 1 and 2.
The arrangements of three things, A, B, C, are in
number six; for AB can form with C, the third, three
different ones BAG, BOA, C B A, and tiiere can
-be no more. Hence it is eyident, that the required
number is, equal to the preceding multiplied by 3, or
to the product of 1, 2, 3.
4. If we add a fourth thing, for instance D, it is
evident, that as each of the preceding arrangements
may be combined with this fourth thing four ways^ the
ttbove number 6, must be multiplied by.4, to obtain
that of the arrangements resulting from four things ;
that is to say, the number will be 34, or the product of
1, 2, 3, 4.
It is needless to enlarge further on this sulject ; fiwr
it may be easily seen, that whatever be the number of
the things ^iven, the number of the arrangements they
are susceptible of may be found, by multiplying tog^
ther as many terms of the natural arithmetical pro>
gression as there are things proposed.
The following table will shew the immense number
of permutations, or different arrangements, of which
only 12 things are susceptible. We shall afterwards
give the result of the permutations arising from the
twenty-four letters of the alphabet.
IN ABITHMSriC. 53^
Komber ofthiogs, Fennutations.
1 1
2 2
3 6
4 .24
5 120
6. 720
7 6040
8 40320
9 362880
10 3628800
11 .... 39916800
12 479001600
Let it be required to assign what space would contain
all the permutations of the twenty-four letters of the
alphabet f supposing each of them to occupy a square
fine.
> If we first suppose that each letter occupies a square
lineya square inch will contain 144 letters; if we then
multiply 144 by itself, the product 20736 will be the
number of letters which can be contained in a square
foot ; and if the last number be multiphed by 9, the
product 186624 will be the number of letters which
might be contained in a square yard.
Now, as a mile is equal to 1760 yards, a square
mile will contain 3097600 square yards; and '^ we
multiply this number by 9, or the square miles in a
league, we shall have 27878400 for the number of square
yards in the square league, and this number multiphed
by 186624 will give 5202778521600 for the number of
letters which could be contained in a square league.
If this last product be multiplied by 21951022,
or the square leagues on the sunace of the earth, al-
lowing the diameter of it to be 7930 miles, we shall
have 114206305788769075200 for the number of let-
ters which, according to the above supposition, could
be cootained in this surface.
. p 3
94 ^ummrnamTs ■
If we now employ the methpd already gifea to find
the number of the permutations of the 24 letters, by
successiyely multiplying all the terms of the a^thme-
tical progression from 1 to U^ we shall have for the
number of these permutations 6204484017332394393
60000, which is aboVe five thousand times greater
than the number of letters that could be contained on
the surface of the earth; and, as each permutation
consists of 24 letters, it thence follows, that to contain
them, a space 120000 times greater would be neces-
sary. In attempting to form an idea of this immense
i^uHiace, the imagination is, as it were, lost; and it
could hardly be believed, that such an extent would be
required, were it not fully demonstrated by calcula-
tion.
PEOBLEM iv.
•
A club of seven persons agreed to dine together every day
suGcessvoely, as long da they could sit down to toMe d^^
feremtly arranged. How many dinners xoould be neees"
sary for that purpose?
It may be easily found, by the rules already giyen^
that the club must dine together 5040 times, before
they would exhaust all the arrangements possible,
which would require above 13 years.
If any word be proposed, such as AMOlEl, and it be
required to know how many different words could be
formed of these four letters, which will give aU the pos-
sible anagrams of that word, we shall find, by multi-^
plying together 1, 0, d, and 4, that they are in number
24, as represented in the following table.
AMOR
MORA
ORAM
RAMO
AMRO
MOAR
ORMA
RAOM
AOMR
MROA
OARM
RMAO
AORM
MRAO
OAMR
RMbA
ARMO
MAOR
OMRA
ROAM
AROM
MARO
OMAR
ROMA
IN CEAXKXB, 55
The number of all the anagrams possible to be formed
of one w6rd^ i!nay be fotitid in the same ixianner ; but it
riltist be confessed, that, if there were a great many
leftters in the ti^ord, the arrangements thence resultiiig
would be so numerotis, as to require a long time to
find them out.
APPLICATION OF THE DOCTRINE OF COM-
V BINATIONS TO GAMES OF CHANCE AND TO
PROBABILITIES.
Though nothing, on the first view, seems more fo-
reign to the province of mathematics, than games oif
chance, the powers of analysis have, as we may say,
enchained this Proteus, and subjected it to calculation :
it has found means to measure the different degrees of
die probability of certain events, and this has given
rise to a new branch of itaathematics, the principles of
which we ^hall here explain.
When ai| eVent can take place in several different
ways, the probabHity of itfe happening in a certain de-
terminate manlier, is greater ^hen, in the whole of the
ways in which it can happen, the greater number of
them determine it to happen in that manner. In a
lottery, for example, every one knows that the proba-
bility of obtaining a prize is greater, according as the
number of the prizes is greater, and as the whole num-
ber of the tickets is less. Tbe probability of an event^
therefore, is in the compound ratio of the number of
cases in which it can happen taken directly, and of the
total number of those in which it can be varied, taken
inversely ; consequently, it may be expressed by a frac-
tion, having for its numerator the number of the favour-
able chances, and for its denominator the whole of the
chances.
Thus, in a lottery, contHiiiiing 1000 tickets, 25 of
which only are prizes, the probability of obtaining a
prize will be represented by tSIttj or ^; if there were
50 pritses, ihe probability would be double; for in that
case it would be equal to •^; but if the number of
tickets, instead of 1000, were 2000, th^ probability
b4
56 AMV8ElffiNT9
would be only one half of the former, or ^; and if
the whole number of the tickets were infinitely great,
the prizes remaining the szme, it would be infinitely
tmall, or ; while, on the other hand, it would become
certainty, and be expressed by unity, if the number of
the prizes were equal to that of the tickets.
Another principle of this theory, the truth of which
may be readily perceived, and which it is necessary here
-to explain, is as follows :
A person plays an equal game when the money
staked, or risked, is in the direct ratio of the proba*
bility of winning ; for to play an equal game, is nothing
else than to deposit a sum so proportioned to the pror
bability of iidnmng,, that, after a great many throws,,
the player may fiikl himself neariy at par ; but for this*
Eurpose,,the stakes must be proportioned to the proba*
ility which each of the [layers nas in his favour. Let
us suppose,, for example, that A bets against 6 on ai
throw of the dice, and that there are two chances in
favour of the former,, and one for the latter: the game
will be equal, if, ai^r a great number of throws, they
separate nearly without any loss. But, as there are
two chances in favour of A, and only one for 6, after
300 throws A will have won nearly 200, and B 100.
A, therefore, ought to deposit 2 and B only 1 ; for by
these means A in winning 200 throws, will get 200;
and B, in winning 100, will get 200 also. In such
cases, therefore,, it is said,, that there is two to one in
favour of A. ,
PROBLEM r.
In tossing tip, what probability is there of throwing a htai
several times successively, or a tail; or, in playing with
several pieces, what probability is there that they will all
come up heads at one throw ?
As this game is well known, it is needless h&te, to
give an explanation of it ; we shaU therefore proceed ta
anfjyze the problem*
IN CHANCES. 57
Ist. It is evident, that as there is no reason why a
head should come up rather than a tail, or a tail than a
head) the probability of one of them coming up is equal
to }, or an equal bet may be taken on either side.
But, if any one should bet to bring heads successively
in two dirows, to know what in this case ought to be
staked on each side, we must observe, that all the
combinations possible of head and tail, which can take
place, in two successive throws with the same piece^
are head, head; head, tail; tail, head; tail, tail; one of
which only gives head, head. Here then there is only
one case in four favourable to the person who bets to
throw a head twice in succession; the probability, .
therefore, of this event will be only | ; and he who bets
that it will take {dace, ought to deposit only a crown,
while his antagonist deposits three : for the latter has
three chances of winning, whereas the former has only
one. To play an equal game, the money deposited by
each ought to be in this proportion.
It will be found, in like manner, that he who should
bet to bring a head, for example, three times succes-
sively, would have in his favour only one of the eight
combinations of head and tail, which might result from
three successive throws of the same piece. The pro-
bability, therefore, of this event would be -J, while that
of his adversary would be |., and consequently, to play
an equal game, he ought to bet only 1 to 7.
It is needless to go over all the other cases; for it
may be readily seen that the probability of throwing a
head four times successively would be only -j-'^, and so
on. We shall say nothing farther, therefore, on the
dififerent combinations which might result from head
and tail ; as in all such cases the following general rule
may be employed.
When the probability of two. or more individual
events are known, the probability of their taking place
all together may be found, by multiplying together the
probabilities of those events considered individually.
Thus* the probability of throwing a head, considered
individually, being expressed at each throw b^ \^ VIcl^X.
of throwing it twice successively, "wiilW)^ ^ X \^ \»
D 5
58 ABTOSBBIEMTS
that of throwing it three times successively, will be
J X J X 1=^, and so on.
2nd. The probability of throwing all heads, or all
tailSf with two, three, or four pieces, may be determined
in the same manner. When two pieces are employed,
there are four combinations of head and tail, only one
of which is both heads ; when three pieces are tossed
up, at the same time, there are 8, one of which only is
au heads; and so on. The probability, therefore, in
each of these cases, is similar to those already ex-
amined.
It may be seen, indeed, without the help of analysis,
that these two questions are absolutely the same, as
may be proved in the following manner: — ^To toss up
the two pieces, A and B, at the same time, pr to throw
up the one after the other, when A the first has had
time to settle, is ceitainly the same thing. Let us
suppose, then, that when A the first has settled, in-
stead of tossing up B, the second. A, is taken from the
ground, in order to be tossed up a second time : this
will certainly be the same thing as if the pfece B had
been employed ; for, by the supposition they are both
equal and similar, at least in regard to the chance of a
head or a tail coming uppermost-; consequently, to toss
up the two pieces A, B, at once, or to toss up twice
successively the piece A, is the same thing.
3rd. If it were asked, how much a person might bet
to bring a head at least once in two throws, it may be
found by the above method that the chance is 3 to 1.
In two throws there are four oombineltions, three of
which give at least one head, while there is only one
which gives two tails; and hence it follows, that there
/are three combinations in favour of the person who
bets to bring a head once in two throws, and only one
against him.
PROBLEM II.
Any number of dice being given, to determine the probability
of throwing with them an assigned number of points.
We here suppose that the dice are of the usual kind,
that is to say, having six faces marked with the num-
«
bers 1, 2, 3, 4, 5, 6. This being premised, we shall
analyse some of the first cases of die problem, th^t we
may proceed gradually to those which are more compleit.
1st. It is proposed to throw a determnate pointy for
example f 6, xffith one die.
As the die has six faces, one of which only is miarked
six, and as any one of these may come up as readily as
another, it is evident that there are 5 chances against
the person who undertakes to throw 6 at one tiirow,
and only 1 in his favour. To play an equal game he
ought, therefore, to bet no more than 1 to 5.
2nd. Let it be proposed to throw the same point 6,
with two dice.
To analyze this case, it must first be observed that two
dice give 36 di£ferent combinations ; for each of the
faces of the die A, for example, may combine with
each of those of B, which will produce 36 combina-
tions. We must next examine in kow many ways the
point 6 can be thrown with two dice. Ist. It will be
found that it can be thrown by 3 and 3; secondly, by
throwing 2 with the die A, and 4 with the die B, or 4
with A and 2 with B, which, as may be readily seen,
forms two distinct cases ; thirdly, by throwing 1 with
A and 6 with B, or 1 with B and 5 with A, which
likewise forms two cases : these are evidently all the
ways that can be found. Consequently, there are 5
cases favourable in the 36; and, tlierefore, the proba-
bility of throwing 6 with two dice, is y*^, and that of
not throwing it ^; hence it* appears that the money
staked by the players ought to be in the ratio of these
two fractions.
By analysing the other cases it will be found, that
of throwing 2 with two dice, there, is 1 chance in 36;
of throwing 3, there are 2; of throwing 4, there are 3;
of throwing 5, there are 4 ; of throwing 6, there are 5 ;
of throwing 7, there are 6 ; of throwing 8, there are 5 ;
of throwing 9, there are 4 ; of throwing 10, tKere are 3;
of throwing 11, there are 2j and of throwing 12, or
sixes, there is 1.
If three dice were proposed, with which the least
point that could be thrown is evideu\i^ %y %sA ^<t
d6
60 AMUSEMENTS
greatest 19^ it will be found, by a similar analysis, tkat
in 216 different throws possible, with three dice, there
is on6 chance of throwing 3-; three of throwing 4 ; six
of throwing 5 ; and so on^ as may be seen in the an-
nexed table, the use of which is as follows.
If it be required, for examjde, in how many different
ways 13 can be thrown with three dice; look in the
first vertical column, on the left, for the number 13, and
at the top of the table for that indicating the number
of the dice, and in the common square of both, opposite
to 13, will be found 21, or the number of ways in
which 13 can be thrown with 3 dice. It will be found,
in like manner, that with 4 dice it may be thrown 140
ways; with 5 dice^^ 420 ways; and so on.
\
n CHANCES.
A TABLE
0/tke dijh-ent waya in tckich am/ point can be
tirown ■ailh one, ttvo, three, or more dice.
i
£
a
I. n.^
III.
THE Diet. 1
IV. V. Vi. 1
1
' 1 1
'2
1
I
3
1
'■^
1
4
1
3
3
1
5, 1 1 4
6j 1 5
10
4
~To
1
5
1
7\ 1 6
15
w
15
(
8| , 5
21
35
35
21
y
1 '
5b
70
5(
1 3
27
HO
12ti
13f
11
1 2
57
104
205
252
12 1 1
25
125
305
45(
13, 1
21
140
420
75t
14
1 1 15
14b
540
lltil
15
10
140
asi
\tim
lb
6
125
735
i2i1
17
3
104
780
285f
IW
1
BO
780
3431
ly
5fa
735
390f
20
35
651
4221
■il
1
20
540
420
4332
4221
2'2
10
aa
4
305
3y0(j
24
1
205
126
34a I
2856
25
Wh^n once it is known in how many ways any point
can be thrown with a certain number of dice, it will be
easy to determine the probability of throwing it. No-
thing will be necessary but to form a fraction, haying
for its numerator the number of ways in which the
point can be thrown , and for its denominator the num-
ber 6, raised to that power denoted by the number of
dice; for example, the cube of 6, or 216, for 3 dice;
the biquadrate, or 1296, for 4 ; and so on.
Thus, the probability of throwing 13 with 3 dice is
•^; that of throwing it with 4 is iV^-
Various other questions of the like kind might be
proposed, some of which we shall here analyze.
PROBLEM III.
JFhen two persons are playing^ to determine the advantage
or disadvantage on the side of the one who undertakes
to throw a certain face, for example , that marked 6, in
a certain number of throws.
Let us first suppose that the person undertakes it at
one throw. To determine the probability of his suc-
ceeding, we must consider, that he who holds the die
has only one chance of winning, and that there are five
of his losing; consequently, to undertake it at one
throw, he ought to bet only I to 5. There is, there-
fore, great disadvantage in undertaking on an even bet
to throw 6 at one throw.
To determine the probability of throwing at least
one faice marked 6, in two throws, with the same die,
we must observe, as has been already said in regard to
tossiiig up, that this is the same thing as to undertake
to bring one face marked 6, by throwmg two dice at
the sdme time. In this case, he who holds the dice has
only 11 chances, or combinations, in his favour; for
he may bring 6 with the first die, and 1, 2, 3, 4, or 5,
with the second; or 6 with the second die, and 1, 2, 3,
4, or 5, with the first; or 6 with each die. But there
are 25 chances, or combinations, against his winning, as
maybe seen in the following table:
IN CHANCES. 6S
I'.l ^ • •! 3 • .1 4«.l 5 • •!
1..2 2«*2 3*.2 4«.2 5*«2
1..3 2..3 3«.3 4..3 5«*3
1..4 2*.4 3..4 4 «4 5*.4
1*«5 2 • • 5 3 • • 5 4 • • 5 5 • • 5
Hence it may readily be concluded, that he who un-
dertakes to throw a 6 with two dice, ought to bet only
11 to 25 ; and consequently, that there is a disadvan-
tage in. undertaking it on an even bet.
We must here observe, that 36, or the whole of
the chances possible with one throw of two dice, is the
square of the given number 6, the number of tlie faces
of one die; and that 25, the number of the chances
unfavourable to the person who undertakes to throw a .
determinate face, is the square of the number 6 di-
minished by unity, that is of 5 ; for this reason, the
number of the favourable chances in the present case,
is the difference of the squares of 36 and 25, or of the
-"square of. the number of the faces of the die, and of
that of the faces of the same die less 1.
To determine the probability of throwing a 6 in three
throws of the same die ; we must, in like manner, con-
sider, that this is the same thing as to undertake to
bring at least one 6 by throwing three dice ; but of the
216 different combinations, produced by 3 dice, there
are 125 in which there is no 6, and 91 where there is
at least one 6 : consequently, he who bets on throwing
one, 6, either in three throws with one die, or in one
throw with three dice, ought to bet no more than 91
to 125; audit would be disadvantageous to undertake
it on an equal bet.
We must again observe, that 91 is the difference of
the cube of the number of the faces of on^ die, viz. 216,
and of 125, the cube of the same number diminished by
unity, that is to say of 5. Hence it may seen, that to
determine, in general, the probability of throwing any
assigned face in a certain number of throws, or in one
throw with a certain number of dice, we must raise 6,
the number of the faces of one die, to that ^>R^t vcl-
dicated by the number of throws gWeu, ot ol ^<& &<i^
64 AMUSEMENTS
to V^ thrown at once, and that we must then ncise to
the same power 6 less unity, that is to say 5, and sub-
tjract it from the former : the remainder, and this power of
5, will be the respective number of chances for winning
or losing.
For example, if a person should bet to bring at least
one 3 with four dice ; we must raise 6 to the biquadrate
or fourth power, which is 1296, and subtract from it
625y which is the fourth power of 5 : the remainder
671 will be the number of chances favourable for win-
ning; and 625 will be those of losing: consequently,
there will be an advantage in laying an even bet.
There will be still more advantage in undertaking,
on an even bet, to throw a determinate point, for ex-
ample 3, in five throws, or with five dice; for if we
deduct the fifth power of 5, which is 3125, from 7776,
the fifth power of 6, the remainder 4651 will be the
number of the favourable chances, and 3125 that of
the unfavourable.
Consequently, to play an equal game, he who bets
ought' to deposit 4652 to 3125, or about 3 to 2.
PROBLEM IV.
In how many thrmos may a person^ with an equal chemce
of winning, bet to bring a determinate doublet , for tx^
ample, sixes, with two dice 1
We already know that the probability of not throw-
ing sixes with two dice is f^ ; consequently, the pro-
bability of not throwing them, in two throws, will be as
the square of that fraction; in three throws, as the
cube; and so on. But as the powers of any number
ever so little greater than imity, go on always increas-
ing, those of a number ever so little less go on always
decreasing; consequently, the consecutive powers of
4^ will go on always decreasing. Let us conceive 4^
raised to such a power, that it shall be equal to | ;
now it will be found that the twenty-fourth power of
ai^tt Uttle greater than f ; and that the twenty-fifth
IN CHANCes. 6$
power of the same fraction is a little less than ^; a
person may then lay an even bet, with some advan-
tage, that another will not bring sixes in 24 throws,
but an even bet cannot be laid with advantage that
sizea will not come up in 25 throws. Consequently,
there is a disadvantage in laying an even bet to bring
sixes in 24 throws ; and, on the other hand, he who
lays an even bet to throw sixes in 25 throws, does so
with advantage.
PROBLEM V.
What probability is there of throwing a determinate daU"
hletyfor example, two threes , at one throw, with two or
more dice ?
To determine the probability in this case, we must
consider, that in undertaking to throw two threes with
2 dice, there is only 1 favourable chance in the 36
^ven by 2 dice. Whence it follows, that the person
who undertakes it, ought to bet only 1 to 35 ; if 3
dice were proposed, we should find that the bet ouebt
to be 16 to 216 ; for the number of chances or conu)i-
nations possible with 3 dice is 216; but when it is
required to throw two threes with 3 dice, it may be
4one 16 different ways; for of the 36 combinations of
the dice A and 6, all those in which there is only one
3, as 1, 3; 3, 1 ; &c. which are 10 in number, by *
combining with the face marked 3 of the die C, Will
consequently give two threes. Besides, the combina-
tion 3, 3 of the dice A, B, by, combining with one
of the six faces of the third C, will also give two
threes ; and hence there are 16 different ways of throw-
ing two threes with 3 dice, which gives 16 favourable
chances in 216. Consequently, the probability of
Growing two threes with 3 dice, is -^ ; and there-
fore the person who undertakes it, ought to bet no
piore than 16 to 216, or nearly 2 to 27.
If the probability of throwing two threes with 4 dice
be required, we shall find that it is expressed by t^^;
$6 AmjmMssrs
for, <^the 1296 combinations arisingfrom the faces of
4dioey there are 150 t^hich give 2 threes^ 20 which
gife 3 threes, and 1 which gives 4, making altogether
171 throwiB, in which there are either 2, 3, or 4uiTee8.
Ofkisequentity, a person ought to bet no more than 19
to 144, or about 1 to 7|, on throwing at least 2 threes
with 4 dice.
In the last place, if the probability of throwii^ any
doublet with ten dice, or more, at one throw, be re-
quired, it will be easy to determine it by the same me-
tiiod of calculation. For, in the case of an indetermi-
nate doublet, it is evident that the probability is six
times as great as when a particular doublet is as-
signed; and therefore nothing will be necessary bnt to'
multiply the above probabilities by 6. The probabi-
lity therefore with 2 dice, is ^ or •^; with 3 mce^-f^
= ^; with 4 dice H^ =i |^; so that there is an ad-
vantage in laying an «ven bet, t<o bring at least one
doublet with 4 dice.
PROBLEM VI.
TxDO persons deposit a certain sum of money ^ and agree,
that he who Jirst gets a certain number cfgames^ for
example 3, shall have the whole ; one of them has got
tXDO games f and the other one; but being untoilUng to
continue their play, they resolve to divide the stake, in
what manner must this be done ?
. This problem is one of the first which engaged' the
attention of Pascal, when he began to study the calcu-
lation of probalnlities. He proposed it to M. de Fer-
mat, a celebrated geometrician of that period, who
resolved it by a different method, viz. that of combi-
nations. We shall here give both.
It is evident that each of the players, when he de-
posited his money, resigned all right to it ; but, on the
other hand, each had a right to that which chance
might give him; consequently, when they give over
playing, the stake ought to be divided according to the
probability each nad of winning.
IN CHANCES. 67
, CASE I.
This proportioti may be determined by the follawini;
mode of reasoning. Since the first player wants one
game to be out, and the second two, it may be readily ^
perceived 9 that if they continued their play, and if the
second won one game, he would want, in the same
mttnner as the first, one game to be out; and if both
players were equally advanced, their hopes of gaining
the >^hole would be equal : in this supposition, there-
fore, they would have an equal right, to the stake, and
consequently, each ought to have an equal share of it.
It is evident, therefore, that if the first wins the game'
about to be played, the whole stake will belong to nim;
and if he loses it, he will be entitled onty to one half. As
the one case is . as probable as the other, the first has
a Hght to the half of these sums taken together ; but
together they make f , the half of which is |. Such is
the portion of the stake belonging to the first plaver,
and, consequently, that belonging to the second is
only|.
CASE II.
The solution of the first case will enable us to re-
solve the second, in which Y^e suppose, that the first
player wants one game to be out, and the second
three ; for if the first should win one game, the whole
of the stake would belong to him, and if he should lose
one, to that the second should want only two games
to be out, I of the money would belong to the former,
since they would then be in the situation alluded to in
the first case. But as both these events are equally
probable, the first ought to have tlte half of these two
sums taken together, or the half of 1, that is to say ^:
the remainder -J? ^U be what ought to belong to the
second.
CASE III.
*
It will be found, by reasoning tn the same manner,
if we suppose two games wanting to the first ^la.^^v
68 AMITSEMENTr
and three^to the second, that on ceasing to playtiiej
ought to divide the stake in such a manner, tfaAt tbe
firit may have -f^y &nd the second ■^,
CASE IV.
Ijf they had agreed to play four games, and if the
first wanted only two games, while the second wanted
four, the stake ought to be divided in such a manner,
that the first might have -fl, and the second •^.
We shall now explain the second method of resolv-
ing questions of this kind, which is that of combina-
tions.
To resolve, for example, the fourth case, in which
we suppose that the first player wants two games to
be out, and the second four, so that both together
want six games; if we subtract unity from' that sum^
we shall have 5, which indicates, that we must take
these five similar letters aaaaa, favourable to the first
player, and the five following, bbbbb, favourable to the
second. These must be combined together, as seen in
the following table, where, of 32 combinations, the
first 26, towards the left, where a occurs at least twice,
indicate the number of chances favourable to the first,
and the 6 last, towards the right, where a is found at
most only once, indicate the number of chances fa**
vourable to the second.
aaaaa a a a b b a a b h b a b b b h
a a a a b a a b b a a b b b a b b b h a
a a a b a a b b a a b b b a a b a b b h
a a b a a b b a a a a b a b b b b a b b
a b a a a a a b a b a b b a b b b b a i
b a a a a a b a a b b h a a b b b b h h
baaabbaabb
b a a b a b a b b a
b a b a a b b a b a
a b a b a b a b a b
Thus, the hope of the first player will be to that of
ihe second, as 26 to 6^ or as 13 tg 3,
IN CHANCXS. W
To resolve the case in which we suppose that one of
the players has won three games, and the other none:
as he will be the winner who soonest gets four games,
he must take unity from 5, the number of games
wanting to both, which will give 4, and then examine
in how many ways the letters a and b can be com-
bined, four and four. These ways are in number 16,
viz.
a a a a a a h h a h b h b b b b
a a a b a b a b h a b h
cababaahhbab
abaaabbabbba
haaababa
b h a a
But, of these 16 combinations, it is evident there
are 15 in which a is found atieast once; and hence it
appears that there are 15 combinations or chances fa-
vourable to the first player, and only one to the second;
consequently they ought to share the stake in the ratio
of 15 to 1 ; or the first ought to have \^^ and the se-
cond V^
PBOBLEM VII.
A fnoimtehank at a country fair, amused the populace with
thefoUomng game : he had 6 (/ace, each of which was
marked only on one face, the first tvith 1, the second
with 2, and so on to the sixth, which was marked 6 ;
the person who played, gave him a certain sum of money,
and he engaged to return it a hundredfold, if, in thrffW'
ing these six dice, the six marked faces should come up
only once in 20 throws?
Though the proposal of the mountebank does not,
on the first view, appear very disadvantageous to those
who entrusted him with their money, it is certain that
there were a great many chances against them.
It may indeed be seen, that of Uie 46656 combina-
tions of the feces of 6 dice, there is only one which
gives the 6 marked faces uppermost ; the probaiHiity
'TO AuvasMBmss
therefoKe of tbro]smig them, at one throw, is expfiessed
by ts4t7- aQ^ ^ the adyenturer was allowed 20
ithxows, the probability o£ his succeeding was onfy
xvm> which is nearly equal to t^^.* To play an
equfiu game therefore, the mountebank should j^ve en-
^gaged to return 2332 times the mOney deposited.
PROBLEM VIII.
The same menntebank offered a new chance to the person
who had lost^ on the following conditions: to deposit a
sum equal to the former ^ and to receive both the stakes
in case he should bring all the blank faces, in 3 «im;-
cessive throws.
Those unacquainted with the method to be pursued
in order to resolve such problems, are liable to reason
in an erroneous manner respecting dice of this kind;
for, observing that there are five times as many blauk
as marked faces, they thence conclude that it i& 5 to
1 that the person who throws them will not bring any
point. They are^ however, mistaken, as the probability,
on the contrary, is 2 to 1 that they will not come up
all blank.
If we take only one die, it is 5 to 1 that the person
who holds it will throw a blank ; but if we add a se-
cond die, it may be readily seen, that the marked
face of the first may combine with each of the blank
faces of the second, and the marked face of the second
with each of the blank faces of the first ; and, in the
last place, the marked face of the one with the marked
face of the other: consequently, of the 36 combinations
of the faces of these two dice, there are 11, in which
there is at least one marked face. But, as we have
already observed, this number 11 is the difference of
the square of 6, the number of the faces of one die,
and of the square of the same number diminished by
unity, that is to say of 5.
If a third die be added, we shall find, by the like
analysis, that, of the 216 combinations of three dice,
them are 91 in which there is at least one marked
f I
»
face ; and 91 is the difference of the cube of 6 or 216,
ajid the cube of 5 or 125; the result wiU be the same
in regard to the more complex cases ; and hence we
may conclude, that of the 46656 combinations of the
faces of the 6 dice in question, there will be 31031 in
whicb there is at least one marked face, and 15625 in
which all the faces are blank; consequently, the chance
is 2 to 1 that some point, at least, will be thrown ;
whereas, by the above reasoning, it would appear that
5 to 1 might be betted on the contrary being the
case.
PROBLEM IX.
In hffw many throws, with six dice, marked on all their
faces, may a person engage, for an even bet, to throw
1,2,3,4,5,6?
We have just seen that there are 46655 chances to
1 that a person will not throw these 6 points with dice
marked only on one of their faces ; but the case is very
different with 6 dice marked on all their faces ; and to
prove it, we need only to observe, that the point 1, for
example, may be thrown by each of the dice, as well as
the 2, 3, &c. which renders the probability of these six
points, 1,1,3, &c. coming up, much greater.
But to analyze the problem more accurately, we
shall observe, that there are 2 ways of throwing 1, 2,
with 2 dice ; viz. 1 with the die A, and 2 with the die
B ; or 1 with the die B, and 2 with A. If it were
proposed to throw 1, 2, 3, with 3 dice; of the whole
of Uie combinations of the faces of 3 dice, there are 6
which give the points 1 , 2, 3 ; for 1 may be thrown
with the die A, 2 with B, and 3 with C; or 1 with A,
2 with C, and S with B ; or 1 with B, 2 with A, and 3
with C ; or 1 with B, 2 with C, and 3 with A ; or 1
with C, 2 with A, and 3 with B ; or 1 with C, 2 with B,
and 3 with A.
It hence appears, that to find the number of ways in
which 1, 2, 3, can be thrown with 3 dice, 1, 2, 3 must
be multiplied together. In like manner, to find the
T2 AMtmeM£NM
munber of ways in which 1, 2, 3, 4 can be thrown with
4 dice, we must multiply together 1, 2, 3, 4, which
will give 24; and, in the last place, to find in how
many ways 1, 2, 3, 4, 5, 6 can be thrown with 6 dice,
we must multiply together these six numbers, the prch
duct of which will be 720.
If the number ^6656, which is the combinations of
the faces of 6 dice, be divided by 720, we shall have
64f for the chances to I, that these points will not
come up at one throw; and, consequently, a {person
may undertake for an even bet to bring them in 64
throws.
In the last place, as the dice may be thrown 130
times, and more, in a quarter of an hour, a person
may, with advantage, bet more than 2 to 1, that they
will come up in the course of that time.
He who engages for an even bet to throw these
points, in a quarter of an hour, undertakes what is
highly advantageous to himself, and equally disadvan-
tageous to his adversary.
ARITHMETICAL AMUSEMENTS IN DIVINA-
TION AND COMBINATION.
PROBLEM I.
To tell the number thought of by a person.
Desire the person, who has thought of a number, to
triple it, and to take the exact half of that triple if it
be even, or the greater half if it be odd. Then desire
him to triple that half, and ask him how many times .
it contains 9 ; for the number thought, if even, will
contain twice as many units as it does nines, and one
more if -it be odd.
Thus, if 5 has been the number thought of, its tri-
ple will be 15, which cannot be divided by 2 without a
remainder. The greater half of 15 is 8; and if this,
half be multiplied by 3, we shall have 24, which con-
tains 9 twice ; the number thought of will therefore be
4 plus 1 , that is to say 5.
IN CHANCES. 73
II. Bid the person multiply the number thought of
by itself; then desire him to add unity to the number
thought of, and to multiply it also by itself; in the
last' place, ask him to tell the difference of these two
products, which will certainly be an odd number, and
the least half of it will be the number required.
Let the number thought of, for example, be 10, which
multiplied by itself gives 100; in the next place, 10 in-
creased by 1 is 11, which multiplied by itself makes
121; and the difference of these two squares is 21, the
least half of which being 10, is the number thought of.
This operation might be varied by desiring the person
to multiply the second number by itself, after it has
been diminished by unity. In this case, the number
thought of will be equal to the greater half of the dif-
ference of the two squares.
Thus, in the preceding example, the square of the
number thought of is 100, and that of the same num-
ber less unity is 81: the difference Of these is 19, the
greater half of which, or 10, is the number thought of.
III. Bid "the person take 1 from the number thought
of, and then double the remainder; desire him to take
1 from this double, and to add to it the number thought
of: in the last place, ask him the number arising from
this addition, and if you add 3 to it, the third of the
sum will be the number thought of.
The application of this rule is so easy that it is need-
less to illustrate it by an example.
IV. Desire the person to add 1 to the triple of the
number thought of, and to multiply the sum by 3; then
bid him add to this product the number thought of,
and the result will be a sum, from which if 3 be sub-
tracted, the remainder will be decuple of the number
required. If 3 therefore be taken from the last sum,
and if the cipher on the right be cut off from the re-
mainder, the other figure will indicate the number
sought.
Let the number thought of be 6, the triple of which
it 18; and if unity be added it mak^s 19; the triple
74 AMDBEME19TS
of this last immber is 57, and if 6 be added it makes
63, from which if 3 be subtracted the remainder will
be 60 : now if the cipher on the right be cut off, tfie
remaining figure 6 will be the number required.
V. Another method of telling the number any one has
thought of.
These operations, by which a person seems to guess
the thoughts of another, may be introduced very oppor-
tunely in company, when any one asserts that all
amusing tricks are performed by slight of hand. The
following method may be found in Ozanam, but we
have here made some additions to it. 1st. Desire any
person to think of a number, but that we may not
speak in too abstract a manner, it will be best to desire
him to thjnk of a certain number of guineas. 2d.
Tell the person that some one of the company lends
him a similar sum, and request him to add them toge-
ther, that the amount may be known. It will here be
proper to name the person who lends him a number of
guineas equal to the number thought of, and to beg the
one who makes the calculation to do it with great care,
as he may readily fall into an error, especially the first
time. 3rd. Then say to the person, I do not lend you,
but give you 10, add them to the former sum. 4th.
Continue in this manner: — Give the half to the poor,
and retain in your memory the other half. 5th. Then
add: — Return to the gentleman, or lady, what you
borrowed, and remember that the sum lent you was
exactly equal to the number you thought of. 6th. Ask
the person if he knows exactly what remains ; he will
answer Yes : you must then say, And I know also the
number that remains, it is equal to what I am going to
conceal in my hand. 7 th. rut into one of your hands
5 pieces of money, and desire the person to tell how
many you have got. He will reply 5 : upon which
open your hand, and shew him the 5 pieces. You may
then say — I well knew that your result was 5; but if
you had thought of a very large number, for example,
two or three millions, the iresult would have been much
r
greater, and I should npt have been able to put into
my band a number of pieces equal to the remainder.
The person t^en supposing that the re«uU of the cal-
culation must be dinerent, according to the difference
of the number thought of, will imagine that it is neces-
sary to know the last number, in order to guess the
result; but this idea is false; for in the case which we
have here supposed, whatever be the number thought
of, the remainder must always be 5, The reason of
this is as follows: — The sum, the half of which is given
to the poor, is nothing else than tmce the number
thought of plus 10; and when the poor have receiTed
their part, there remains only the nun^r thought of
phis 5 ; but the number thought of is cut off when the
sum borrohved is returned, ajad consequently there re-
msuns only 5.
it may be thence seen, that the result may be easUy
toown, since it will be the half of the number given in
the third part of the operation ; for example, whatei^er
be the number thought of, the remainder will be 36,
or 95, according as 72 or 50 have been given.
Remark 1st, If this trick be performed several times
successively, the number given in the third past of tte
operation must be always different, for if the resu)t
were several times the same, the deception might be
discov^ed.
2nd. When the five first parts of t)ie calculation for
obtaining a result are finished, it will be best not to
name it at first, but to continue the operation to render
it more complex, by saying, for exaraiple, Double the
remainder, deduct two, add three, take the foui^h part,
&c. a^d the different steps of the calculation may be
kept in mind in order to know how much the first re*
suit has been increased or diminished. — This irregular'
process never fails to confound those who attempt to
foUow it.
£2
IN CHANCES. 77
fourth ; and so on to the last ; and then the sum of the
first and the last. Having written down all these sums
in order, add together all those, the places of which
ai^e odd, as the first, the third, the fifth, &c.; make
another sum of all those, the places of which are even,
as the second, the fourth, the sixth, &c. subtract this
sum from the former, and the remainder will be the
double of the first number. Let us suppose, for ex-
ample, that the 5 following numbers are thought of,
VIZ. 3, 7, 13, 17, 20, which when added, two and
two as above, give 10, 20, 30, 37,23; the sum of the
first, third, and fifth is 63, and that of the second and-
fourth is 57 : if 51 be subtracted from 63, the re-
mainder 6 will be the double of the first number 3.
Now if 3 be taken from 10, the first of the sums, the
remainder 7 will be the second number; and by pro-
ceeding in the same manner, we may find all the rest.
In ^he second case, that is to say, if the number of
the numbers thought of be even, you must ask and
write down as above the sum of the first and the se-*
c6nd; that of the second and third ; and so on, as before ;
but instead of the sum of the first and the last, you must
take that of the second and last; then add together
those which stand in the even places, and form them
into a new sum apart; add also those in the odd
places, the first excepted, and subtract this sum from
the former : the remainder will be the double of the
second number ; and if the second number, thus found,
be subtracted from the sum of the first and second, you
will have the first number ; if it be taken from that of
the second and third, it will give the third; and so of
the rest. Let the numbers thought of be, for example,
3, 7, 13, 17: the sums formed as above, are 10, 20,
30, 24 ; the sum of the second and fourth is 44, from
which if 30, the third, be subtracted, the remainder
will be 1 4, the double of 7 the second number. The
fir$t, therefore, is 3, the third 13, and the fourth 17.
£ 3
%
t8 ABfMEBiEI^Tf
FliOBLEM III.
A person havifig in one hand an even number of skiUingSf
and in the other an odd, to tell in which hand he has the
tven number.
Desire the person to mulfiply the number in the right
hapod by any eren number whatever, such as 2 ; and
thart in the left by an odd number, as 3 ; then bid bun
add together the two products, and if the whole sum
be odd, the eten number of shillings will be in the
right hand, and the odd number in the left : if the sum
be even, the contrary will be the case.
I/et us suppose, for example, that the person has 8
shillings in ^ s right hand, and 7 in his left ; multiplied
by 3 gives 10, and 7 multiplied by 3 gives 21 : the sum
of which, 37, is at odd number.
If the number in the right hand were 9, and that in
tie left 8, xVe should have 9x2= 18, and 8 x 3 =
24; the sum of which two products is 42; an even
niumber.
PROBLEM IV.
A person having in one hand a piece of gold, and in the
other a piece of sitter ; to tell in which hand he has the
gold, aid in which the siher.
For this purpose, some value, represented by an even
nnmber, such as 8, must be assigned to the gold, and
a value represented by an odd number, such as 3, must
be assigned to the silver; after which, you may pro-
oeed exactly in the same manner as in the preceding
example.
1st. To conceal the artific<^ better, it will be suflR-
cient to ask whether the sum of the two products can
be halved without a remainder; for in that case the
total will be even, and in the contrary case odd.
2nd. It may be readily seen, that the pieces, instead
of being in the two hands of the same person, may
be Mupposed to be in the hands of two persons, one
IN CHANCEg. 79
of whom has the even number, or piece of gold, and
the other the odd number, or piece of silver. The
same operations may then be performed in regard to
these two persons, as are performed in regard to the
two hands of the same person, calling the one privately
the right, and the other the left.
PROBLEM V.
The game of the ring*
This game is nothing else than an application ot
one of the methods employed to tell several numbets
thought of, and ought to be performed in a company
not exceeding 9, in order that it may be less complex.
Desire any one of the company to take a ring, and to
put it on any joint of whatever finger he may think
proper. The question then is, to tell what person hat
the ring, and on what hand, what finger, and what
joint.
For this purpose, you must call the first person J,
the seevnd 2, the third 3, and so on. You must all^o
denote the 10 fingers of the two hands, by the follow-
ing mumbers of the natural progression, 1, 2, 3, 4, 5,
dEC« beginning at the thumb of the right, and ending
at that of the left, that by this order the number of the
finger may at the same time indicate the hand. In
the last place, the joints must be denoted by 1, 2, 3,
be^nning at the porn^ts of the fingers.
To render the solution of this problem more explicit,
let us suppose that the fourth person in the company
has the ring on the sixth finger, that is to say, on the
little finger of the left hand, and on the second joint of
that finger.
Desire some one to double the number expressing
the person which, in this case will give 8 ; bid him
add 5 to this double, and multiply the sum by 5,
which will make 65; then tell him to add to this pro-
duct the number denoting the finger, that is to say 6,
by which means you will have 71 ; and, in the last
place, desire him to multiply ihe Wluuin^ex V}\^^
£ 4
S AMUSEMENTS
and to add to the product the number of the joint 2*
The last result will be 712; if from this number you
deduct 250, the remainder will be 462 ; the first fi-
gure of which, on the left, will denote the person;
the next, the finger, and consequently the hand, and
the last the joint.
It must here be observed, that when the last result
contains a cipher, which would have happened in the
present . example,, had the number of the finger been
10, you must privately subtract, from the figure pre-
ceding the cipher, and assign the value of ten to the
cipher itself.
The same formula, as may be readily conceived,
will answer for all cases whatever.
PROBLEM VI.
To guesi the number of spots on any card which a person
has drawn from a whole pack.
Take a whole pack, consisting of 52 cards, and de-
sire some person in company to draw out any card, at
pleasure, without shewing it. Having assigned to the
different cards their usual value, according to th^
spots, call the knave 11, the queen 12, and the king
13. Then add the spots of the first card to those of
the second ; the last sum to the third ; and so on, al-
ways rejecting 13, and keeping the remainder to add
to. the following card. It may be readily seen that it
is needless to reckon the kings which are counted 13*
If any spots remain at the last card, you must sub-
tract them from 13, and the remainder will indicate,
the card that has been drawn : if 12 remains, it hat
been an ace; but if nothing remains, it has been a
king.
DEMONSTRATION,
Since a complete pack contains 13 cards of each
suit, the values of which are 1, 2, 3, &c. as far as 13»
the sum of all the spots of each of the different suits.
IN CHANCES. 81
will be 7 times 13(21), which is a multiple of 13; con-
sequently the quadruple is also a multiple of 13 : if we
add, the spots of all the cards, always rejecting 13,
the remainder at last must be 0. Hence it is evident,
that if a card, the spots of which are less than 13, be
drawn, the difference between its spots and 13, will be
what is wanting ft complete the number. If, at the
end, then, instead of attaining to 13, we attain only to
10, for example, it is plain, that the card wanting is a
3; and if we attain exactly to 13, the card missing
must be equivalent to 13 ; that is, it must be a king.
PROBLEM YII.
A person hating a certain number of counters in each hand;
to find how many he has altogether.
Desire the person to convey 4, for example, from the
one hand to the other; and then ask him how many
times the less number is contained in the greater ? Let
us suppose that he says the one is the triple of the
other ; in this case multiply 4, the number of counters
conveyed from one hand into the other, by 3, and add*
the same number, which will make 16. In the last
place, from the same number 3, subtract unity, and if
you divide 16 by 2, the remainder, the quotient 8 will
be the number contained in each hand; and conse-
quently the whole number is 16.
Let us now suppose, that when 4 counters are con-
veyed from one hand to the other, the less number is
contained in the greater 2j. times : in this case, we
must, as before, multiply 4 by 2^, which will give 9^;
to which if 4 be added, we shall have 1^, or Y« Then
if unity be taken from 2^, the remainder will be 1^ or
^; by which if Y be divided, the quotient 10 will be
the number of counters in each hand, as may be easily
proved on trial.
E 5 ... J V. »
80 • AaiU8£M£KT«
PROBLEM YIII.
itteral Cards being grven^ to tell which of them a person
has thought qf\
Desire the person to remember the card, and its
place in the pack, counting from t^ie bottom. Then
uJte the cards, and in a dexterous manner, so as not
to be perceived, convey a certain number of them from
the top to the bottom ; and subtract them in your mind
fioih the pack, with the number of which you are ac-
quainted. If the pack, for example, consists of 52
cards, and you have conveyed 8 to the bottom, tell
the person that the card he has thought of will be the
forty -fourth, reckoning from the card the place of
which he is going to name. Thus, if he says it ts
the ninth, you go on counting 9, 10, 11, &c. and
the card he thought of will be exactly the forty-fourth,
as you announced.
PROBLEM IX.
tt&cing spread out on the table 20 cards, arranged two
and tvcoy and desired one or more persons to t him of two f
provided they lie close to each other ^ to tell which cards
they have thought of.
Yon must retain in your memory the fouf following
words, with the arrangement of the letters which com-
pose them :
m
t
s
a
t
t
a
t
I
to
e
m
n
V
e
s
u
I
Collect all the cards into the left hand, two by two,
as they lay om tlie table, and then place them, on^t by
one, in the same order as the preceding letters, taking
care to place the two first as the two m, the two next
as the two t, the two following as the two s, and
so on.
IK CHiNCEIt. , €&
Ask each person in which horizontal row his two
cards are. If he says they are both in the same row,
for example, the third, they will be pointed out by ttit
letters it and n, contained m that row ; if they are in
two different rows, as the first and last, the letters #
and # will indicate the place which they occupy.
PHOBLEM X.
To make alt the cards of the same kind to be found togp"
tkcFf however often the pack ma^ have been cut*
Have in readiness a pack, all the cards of which
arranged in successive order ; that is to say, if it con**'
sist of 52 cards, every 13 must be regularly arranged,
without a duplicate of any one of them. After thej
have been cut as many times as a person may chooiil^
form them into 13 heaps of 4 cards each, widb the co-
loured faces downwards. When this is done, the 4
kings, the 4 queens, the 4 knaves, and so on must ne-
cessarily be together.
PROBLEM XI.
The four indivisible kings*
Take four kings, and place between the third aad
fourth any two common cards whatever, which must
be neatly concealed; then shew the four kings, aod
place the six cards at the bottom of the pads. ; take
one of the kings, and lay it on the top, and put one ^
the common cards into the papk nearly about the mid«
die; do the same with the other, and then show that
there is still one king at the bottom : desire any one tA
cat the pack, and as three of the kinffs were left at
the bottom, l^e four will therefore be found together
in the middle of the pack.
%6
84 AMUSEMENTS
PROBLEM XII.
Two heaps of cards being displayed on a table, to write
on a piece of paper that heap which a person will
choose.
Place in a heap 2 or 3 sevens ; and in another 7
cards. Write on a bit of paper the word seven, and
invert it, that what you have written maybe concealed :
Uien desire any one to choose, and when that is done,
turn up the heap chosen, and prove the truth of your
prediction by shewing what you wrote ; but you must
take care to she]^ only the heap which has been cho-
sen.
PROBLEM XIII.
Several cards being presented in succession to several
persons, that they may each choose one at pleasure ; to
guess that which each has thought of,
Shew as many cards to each person as there are
persons to choose ; that is to say, 3 to each, if there are
3 persons. When the first has thought of one, lay
aside the three cards in which he has made his choice.
Present the same number to the second person, to think
of one, and lay aside the three cards in the like man-
ner. Having done the same in regard to the third
person, arrange all these cards in three rows, with their
faces turned downwards, and then put them together
in order. If you take the 3 first, and present them
successively to the different persons, and do the same
thing with the others, you may easily guess the cards,
by observing, that the card thought *of by each per-
son will have the same place among the cards as At
person has in regard to the other two ; that is to say,
the card thought of by the first person, will be fi^t of
that packet in which he discovered it; that thought
of by the second, will be the second in the packet^
ivhere he recognized it; and that of the third, will be'
the last and in the last packet.
IN CHANCES. 85
The operation is exactly the same when the num-
ber of persons is greater. If, instead of 3, there are
4, or 5 persons, four or five cards Inust be presented
to each.
PROBLEM XIV.
Three cards being presented to three persons^ to guess that
which each has chosen.
As it is necessary that the cards presented should be
distinguished, we shall call the first A, the second B,
and the third C. Let the persons, whom we shall dis-
tinguish by first, second, and third, choose privately
which ever of the cards they think proper, and when
they have made their choice, which is susceptible of six
varieties, give the first person 12 counters, the second
24, and the third 36 : then desire the first person to
add together the half of the counters of the person who
has chosen the card A ; the third of those of the per-
son who has chosen B; and the fourth part of those
of the person who has chosen C; and ask the sum,
which must be either 23 or 24 ; 25 or 27 ; 28 or 29, as
in this following table :
First.
Second.
Third-
Sam!
12
24
36
A
B
C
23
A
C
B
24
B
A
C
25
C
A
B
27
B
C
A
28
C
B
A
29
This table shews, that if the sum is 25, for example,
the first person must have chosen the card B, the se-
cond the card A, and the third the card C; and that,
if it be 28, the first person must have chosen the card
B, the «econd the card C, and the third the card A ;
aod so of the rest.
I
85 AMDil&MfeNTi
PROBLEM XV.
f
To icUtht nunAer of spots on all the bottom cards of s^vt"
ral heaps, arranged on a table*
Arrange each heap of cards in such a manner, that
the spots on the bottom one, added to the cards above
it, may always amount to 12; continue to make as
many heaps as possible, in the manner above pre-
scribed, and place the remaining^ cards on one side.
Then separate in your mind four heaps, and multiply
the heaps which remain, after these are deducted, oy
13; this product, added to the number of cards, will
be that of the spots required. We shall give the solu*
tion of this problem by an analysis in another place.
* l^ROBLEM XVI.
To name all the cards of a pack*
Hav6 a eomplete pack of 52 cards, and arrange them
according to the order of the following words, which
you must retain in your memory:
Ufuis quinque novem famulus sex quatuor duo
Ace five nine knave six four two
Rex septem octo foemina trina decern
King seven eight queen three ten
Besides this first order, you must arrange them also
accordiiig to the order of the colours, spades, hearts,
clubs, dnd diamonds ; so that the 52 cards may be
disposed as follows :
ORDE& OF TH£ CARDS. ,
1 Ace of spades 5 Six of spades
2 Five of hearts 6 Four of heartt
3 Nine of clubs 7 Two of clubs
4 Knave of diamonds 8 King of diamondt
SK CHANCES.
87
9 Seven of spades
10 Eight of hearts
11 Quelen of cluhs
1% Three of diamonds
13 Ten of spades
14 Ace of hearts
15 Five of clubs
16 Nine of diamonds
17 Knave of spades
18 Six of hearts
19 Four of clubs
do Two of diamonds
21 King of spades
22 Seven of hearts
23 Eight of clubs
24 Queen of diamonds
25 Three of spades
. 26 Ten of hearts
27 Ace of clubs
28 Five of diamonds
29 Nine of spades
30 Knave of hearts
31 Six of clubs
32 Four of diamonds
33 Two of spades
34 King of hearts
35 Seven of clubs
36 Fight of diamonds
37 Queen of spades
38 Three of hearts
39 Ten of clubs
40 Ace of diamonds
41 Five of spades
42 Nine of heaVts
43 Knave of clubs
44 Six of diamonds
45 Four of spades
46 Two of hearts
47 King of clubs
48 Seven of diamonds
49 Eight of spades
50 Queen of hearts
51 Three of clubs
52 Ten of diamonds
This order is of such a nature, that, by knowing any
one of the 52 cards, that which follows it may be also
known.
Thus, for example, if it were required to know what
card follows the king of spades, it will be sufficient to
recollect that septem, in the two Latin lines above given,
vldch follows that of rex y denotes that it is a seven;
and as the colonr which follows the spades is heart8>
it is the seven of hearts, and so of the rest.
' Every thing being thns arranged, having retained in
your memory the above words, and the order of the
colours, desire any person to cut the pack as many
times as he chooses ; for it will be easy to namt all
the cards in order, provided you have found means, bt
some dexterous tnanoeuvre, to observe that on6 whicm
is at the top of the pack.
The same arrangement of the cards may bd em-
ployed for various amusements.
88 AMUSEMENTS
\st. To make a person beHeve thai you can distinguish
the cards by their smell.
The pack being disposed in the above order, present
it to any one, that he may choose a card at pleasure ;
open Uie pack at the place where it has been drawn
out, and dexterously observe that which precedes it, by
seeming to smell the place from which it was taken.
It will then be very easy to name it, as it can be only
that which follows in the order already indicated.
2(f. A pack of cards being divided into two parts; to
discover whether the number in each be odd or even.
First, find out whether the last card in the pack be
black or red ; then, on the pack being cut into two
parts, if the card found at the bottom of the upper
division is of the same colour as that at the bottom of
the pack, the two parts which have been separated,
contain each an even number ; on the other hand, if
it be of a different colour, they contain each an odd
number.
3d, To tell the number of spots on several cards which
any person has chosen.
Having presented the pack, that the person may
choose several succeeding cards at pleasure, privately
observe the card which is above those he has chosen ,
and how many he has drawn from the pack ; it will
then be easy to count how many spots they ought to
contain.
For example, if the observed csurd be a nine, and
four cards have been drawn, it may readily be seen
that those drawn must be a knave, equivalent to 10
spots ; a six, a four, and a two. You may then an*
nounce, that the cards, in the persons hand, contain
22 spots.
IN CHANCES. 89
PROBLEM XVIT.
9
Having desired a person to draw four cards from a pack
and to think of one of them, to tell the one he has
thought of.
Suffer the person to draw four cards from the pack
at pleasure, and desire him to think of one of them;
then take these four cards back, and place two of
them at the top and two at the bottom of the pack, ih
a dexterous manner, so as not to be perceived : under
the two last, place any four cards whatever; then
display the lower part of the pack on the table, shew-
ing only 8 or 10 cards, and ask the person whether
the one he thought of be among them. If he says No,
you may be sure that it is one of the two which you
put at the top of the pack; in that case you must
transfer them to the bottom, and then, shewing the
bottom of the pack, say. Is not this your card? If
he replies No, turn aside that card with your third
finger, which you must have previously moistened, and
desire him to draw out his card himself from the bot-
tom of the pack.
If the person should say, that the card he thought
of is among the first shewn to him, dexterously re-
moye the four cards put at the bottom of the pack, in
order that the two, one of which is the card he thought
of, may be the lowermost of the pack, and you may
then either shew him his card or make him draw it out
himself, as above explained.
PROBLEM XVIII.
Three things being privately distributed to three persons^
to guess that which each has got.
Let the three things be a ring, a shilling', and a
glove. Call the ring A, the shilling E, and the gloved ;
and in your own mind distinguish the persons by call-
ing tnem first, second, and third. Then take 24 coim-
ters, and give one of them to the first ^er&oU) Vh^ Vs^
90 AMUSfiMSNTfl
the second, and three to the third. Place the re*
maining 1 8 on the table, and then retire, that the thre^
persons may distribute among themselves the things
proposed, without your observing them. When the
distribution has b^n made, desire the person who has
the ring to take from the 18 remaining counters, aa
many as he has already; the one who has the shil*
ling to take twice as many as he has already;
and the person who has the glove to take four times
as many. By these different combinations, the
counters left can be only 1,2, 3, 5, 6, or 7. When
this is done, you may return, and by the number of
counters left,. you can discover what thing each has
got, by employing the following words :
1 2 3 5 6 7
Parjer Cesar Jadis devint si grand prince*
To make use of these words, you must recollect
what has been already said, viz. that the number of
the counters which remain, can be only 1, 2, 3, 5, 6 or
7, and never 4: you must observe also, that each
syllable contains one of the vowels which we have made
to represent the three things proposed, and that the
above line must be considered as consisting only of
six ^ords : the first syllable of ^ach word must also
be supposed to represent the first person, and the se-
cond syllable the second person. This being compre-
hended, if there remains only one counter you must
employ the first word, or rather the two first syllables
parfer, the first of which, that containing A, shews
that the first person has the ring represented by A ; and
the second syllable, that containing £, shews that the
second person has the shilling, represented by £; from
which you may easily conclude, that the third person
has the glove. If two counters remain, you must
take the second word Char, the first syllable of which,
containing E, will shew that the first person has the
shilltng, represented by £ ; and the second syllable,
contBining A, will indicale that the second person has
IN CHJ^CES. 91
the ring, represented by A ; yop may then easily coti-
clude Uiat the third person has the glove.
PROBLEM XIX.
To teU, by inspecting a watch, at what hour a person ha$
resolved to rise next morning,
1st, When the person has thought of an hour, bid
him touch some other hour on the dial-plate, and then
desire him to add 12 to it privately in his own mind,
which will form a certain number.
2nd. Then desire him to proceed backwards, and to
count the above number, beginning with the hour
which he thought of.
Let the hour thought of, for example, be 8, and that
touched be 3 ; as 12 added to 3 makes 15, desire the per-
son to count that number, in a retrograde order from the
hour inched, beginning with 8, the hour thought of;
counting 8 on the hour 3, 9 on 2, 10 on 1, and so on,
by which means 15 will fall upon the hour of 8.
The person will be surprised to find that he has
fallen on the hour he thought of.
PROBLEM XX.
\
Two persons agree to take alternately numbers less than a
given number, for example 1 1 , and to add them ioge^'
ther till one of them has reached a certain sum, such as
100; by what means can one of them infallibly attain
to that number before the other ?
The whole artifice of this problem consists in imme-
diately making choice of the numbers 1, 12, 23, 34,
and so on, or of a series which continually increases by
11, up to 100.
Let us suppose, that the first person, who knows
the game, makes choice of 1 ; it is evident that His
adversary, as he must count less than 11, can at most
reach 11 by adding 10 to it. The first will then take
1, which wifl make 12 ; and whatevex \wxm\^t >^ \<^-
93 AMUSEMENTfl
coud may add, the first will certainly win, provided he
continually adds the number which forms the comple-'
ment of that of his adversary to 11 ; that is to say, if
the latter takes 8, he must take 3; if 9, he must take
2, and so on. By following this method, he will infal-
libly attain to 89; and it will then be impossible for
the second to prevent him from getting first to 100;
for whatever number the second takes, he can attain
only to 99 ; after which the first may say, And 1 makes
100. If the second takes 1, after 89, it would make
90; and his adversary would finish by saying, And 10
make 100.
It is evident, that when two persons are equally well
acquainted with the game, he who begins must neces-
sarily win.
PROBLEM XXI.
Sixteen counters being disponed in two rowSy to find that
which a person has thought of.
The counters being arranged ad follow, desire the
person to think of one, and to observe well in which
row it is :
A B C D - E F HI
#
#
#
#
00 00 00 00
00 00 00 00
00 00 00 00
Let us suppose that the counter thought of is in the
row A : take up the whole row in the order in which
it now stands., and dispose it in two rows C and D, in
such a manner, that tlie first counter of the row A may
be the first of the row C ; the second of the row A, the
first of the row D; and so on, transferring the 16
IN CHANCES. 93
counters from A and B, to C and D. This being done,
again ask in which of the vertical rows the counter
thought of stands. We shall suppose it to be in C;
remove that row as well as D, observing the same
method as before; and continue in this manner until
the counter thought of becomes the first of the row 1.
If you then ask in which row it is, it may be imme-
diately known, because after the last operation it will
be the first in the row said to contain it ; and as each
row has a distinguishing character or sign, you may
cause them all to be mixed with each other, and stiU
be able to discover it by the sign you have remarked*
Instead of 16 counters, 16 cards may be employed.
After you have discovered the one thought of, you may
cause them to be mixed, which will conceal the arti-
fice.
If a greater number of counters or cards be em-
ployed, disposed in two vertical rows, the counter or
card thought of will not be at the top of the row after
the last transposition: if there are 33 counter^ or
cards, 4 transpositions will be necessary ; if 64, there
must be 5; and so on.
PROBLEM XXII.
A certain number of cards being shown to a person, to
guess that which he has thought of.
To perform this trick, the number of the cards must
be divisible by 3; and to do it with more convenience,
the number must be odd.
The first condition, at least, being supposed, the
cards must be disposed in three heaps with their faces
turned upwards. Having then asked the person in
which heap is the card thought of, place the heaps
one above the other, in such a manner that the one
containing the card thought of may be in the middle.
Arrange the cards again in three heaps, and having
asked in which of them is the card thought of, repeat
the operation as before. Arrange them a third time
in three heaps, and having once more asked the sam^
9i AMUSEMENTS
Siestion, form them all into one heap, that containing
e card thought of being in the middle. The card
thought of must then necessarily be the middle one,
that is to say, if 15 cards have been employed j it wiU
be the eighth from the top; if 21, the eleventh; if 27,
the fourteenth ; and so on. When the number of > the
cards is 24, it will be the twelfth, &c
PEOBLEM XXIII.
To arrange 30 crindnah in suck a manner , that hy counts
ing them in succession, always beginning again at the
first J and rejecting every ninth person y 15 of thtm may
he saved,
Arrangfe the criminals according to the order of the
TOweLs, in the following Latin verse,
4521 81 13 381321
Populeam virgam mater regina J'erebat,
Because is the fourth in the order of the vowels,
you must begin by four of those whom you wish to
save ; next to these place five of those whom you wish
to punish; and so on alternately, according to the
figures which stand over the vowels of the above
Terse. /
In a company consisting of several persons , the following
game may be introduced by way of amusement.
We shall suppose that there are 13 ladies in the
company; in that case, provide 12 nosegays, and in
order to mortify one of them, without shewing any
appearance of partiality, announce that you mean to
let chance decide which of them is to go without one.
For this purpose, make the 1 3 ladies stand up in a
ring, allowing them to place themselves as they [dease ;
and distribute to ihem the 12 nosegays, counting them
from 1 to 9, and making the ninth retire from the ring
and carry with her a nosegay. It will be found, that
the eleventh, reckoning from the one by whom you
IN CHANCES.
95
began, will remain the last ; and consequently will hate
no share in the distribution.
The following table will shew the person, before her
whom you wish to exclude, with whom you must begin
to count 9, supposing always that the number of the
nosegays is less by 1 than that of the persons.
'I For
13 persons, the 11th before. >.
■■•
12 . .
. ... 2d.
<ij»
11 . .
. . . . 5th.
* ^.
10 • .
9 . .
8 .
. . . 7th.
» . . . 8th.
. . . . 8th.
•j?.
7 . <
. . . . 7th.
t
6 . -
. . . . 5th.
^
5.
4 . <
. * • . 3d.
•• .,
3 • .
. ... 2d.
2 .
• • -•
• • . . 1st.
i »,
S r\
■■» T
TVK* ■^y%T t -WT
A man has a wolf, a goat, and a cabbage, to carry over a
river, but, as he is obliged to transport them one by one,
in what manner is this to be done, that the wolf may
not be left with the goat, nor the goat with the cab-
bage ?
He must first carry over the goat, and then return ^
for the wolf; when he carries over the wolf, he must
take back with him the goat, which he must leave, in
order to carry over the cabbage ; he may then return,
and carry over the goat. By these means the wolf
will never be left_with the goat, nor the goat with the
cabbage, but when the boatman is present.
PROBLEM XXV.
Jn what manner can coimters be disposed in the eight ejr-
temal cells of a square^ so that there may always be 9
in each row, and yet the whole nymber shall vary from
20 to 32 ?
This problem may be proposed in the following
manner : — ^A wine merchant caused 32 casks of choice
96
AMUSEMENTS
wine to be deposited in his cellar, giring orders to his
clerk to arrange them in the annexed figure, so that
each external row should contain 9.
1*^ Order.
1
7
1
■ 7 -*
.
' 7
1 7
1
The clerk, however, took away 12 of them, at three
different times; that is, 4 each time; yet when the
merchant went into the cellar, after each theft had been
committed, the clerk always made him count 9 in each
row. How was this possible ?
This problem may be easily solved by inspecting
the following figures :
2rf Order.
Zd Order.
2
5
2
•
3
3
3
5
5
3
•
3
2
5
2
3
3
3
Ath Order.
4
1
4
1
1
4
1
4
vr
T0 distribute atmmg 3 permui, il eaih rftm^f 7 (^thm
fully 7 of them m0^ and 7 rf tkm hajf fM, «o that
each of them shdfnave the same quantity of xme, and
the same number of casks.
This problem admits of two solutions, which may be
clearly eomprehended by means of the two foBowing
taMes:
rerfAM.
FnUCttdu.
Vmpf.
mautdi.
fist
2
2
3
I. ^2d
2
2
3^
(3d
3
3
I
PersoDf.
Full Catks.
Empty.
Half-fall.
fist
3
3
1
11. ^2d
3
3
1
(.3d
1
1
5
(
* PROBLEM
XXVII.
A schoolmaster, to amuse his schotars, shewed them a num*
her, tohich he said was the sum of 6 roxos, each consist-
ing of 4 figures ; he then desired them to xorite down
"3 rows offiguresj to which he would add 3 more, ani
assured them that the sum of the whole should be equal
to the number he shewed them.
To solve this tnroblem, mritiply in your own mind
9999 by 3 ; and the product 29997 will be the num-
ber wldch the schoolmaster shewed to his scholars.
Rowsof thef^to
scholars, \ 3^^^
Rows of the
master
the(
2714
4170
6543
Total 29997
F
98 ABOJSEBfENtS
It may be here seen, that each figure set down by
the master is the complement to 9, of that set down by
the scholars ; and consequently the sum, thoujgfa writ-
ten down beforehand, must be exact.
PROBLEM XXYIII.
Having desired awf person to mukiph/,for example ^ one of
the three foUowing numbers, hy any figure at pleasure^
and to tell you the product , after suppressing one Jigwre
of ity and even changing the order of the rest, to guess
the figure that has been suppressed.
Let the three given numbers be
- 364851
234765
823644
If we suppose the person to multiply the third num-
ber by 6, the product of which will be 4041864, what-
ever figure be effaced, it may be easily discovered by
that wanting to complete the product, as the sum of
its figures must necessarily be a multiple of 9. If the
6, for example, be suppressed, the sum will not be a
multiple of 9 ; for it amounts only to 30:' as 6 there-
fore is wanting to 30 to make it a multiple of 9, you
may boldly assert that 6 has been suppressed.
As the sum of the figures would still be a multiple
of 9, if a cypher were suppressed, and as it would con-
sequently have no need of being complete, you must
make it a condition of the problem that the person shall
suppress only one significant or effective figure; and
if you find that the sum has no need of being com-
pleted, you may conclude that the figure suppressed
has been a 9.
A mountebank, to give the greater air of the mar-
vellous to this sport, pretended to discover by the
smell what figure had been suppressed ; but it may
easily be supposed, that while he pretended to smell
the figures, he privately added them together, so as to
discover their sum.
IN CHANCE8. 99
There is another method of guessing the suppressdtl-
figure, even when the person has been allowed to write
down the sum to be multiplied himself; but in this
case you must stipulate to have permission to add any
one figure you choose : you must observe what figurie
is wanting to complete the sum, and set down that fi-
gure ; if nothing is wanting, you may add 0, or 9.
PROBLEM XXIX.
A person having made choice of two numbers, and mvUi-
plied them together, to tell the product, provided you
know only the last Jigure of it.
Have in readiness a small bag with two divisions,
and put into one of them 12 square bits of card, each
inscribed with the. number 73; and into the second 9
other pieces, inscribed with the terms of the arithme-
tical progression, 3, 6, 9, 12,, 15, 18, 21, 24, 27.
Present that aperture of the bag which contains the
numbers 73, and desire the person to draw out one ;
then dexterously change the side of the bag, and hav-
ing desired another person to draw any number from
the second division, bid him multiply the number he
has taken by that drawn out by the first person: the
product will necessarily be one of the nine numbers
219, 438, 657, 876, 1095, 1314, 1533, 1752, 1971.
You may then easily tell the product of the multipli-
cation, if you know only the last figure of it.
It must here be observed, that this recreation re-
quires a good memory ; as it will be necessary to know
by heart the . above nine products. The following,
founded on the same principle, is much easier.
PROBLEM XXX.
A person hadng chosen two numbers, and divided the
greater by the less, to tell the auotient ; that is to say,
how many times the less is contained in the greater.
Put into the first division of the bag the nine num-
bers, 219, 438, 657, 876, 1095 1314, 1533, 175^,
t2
100 4Mv§tmv!m
1971 ; and into the «eooiid« th^ cards wscrihed with
the numbeir 73. Desici^ t^e perscA to draw a ownber
ffom each diyiaiociit ai|d to divide ^e one by the other ;
then aik him to ^ jfq^ the laat ^ure of the greater
of the nuiabefs, ai^^it; wiU enable joa to diacover whkh
q£ the nijoe nun^bevs ^f the above arithmetical pro-
gpresston is the c^ofi^eoX: tkm, if it be a 9» the nwn-
ber 3 is the quotient; if it be an 8, the quotient is the
number 6; and soon. For the quotient 3, 6, 9, 12,
15, 18, 21, 24, and 27, will be in the ratio of the
figures 1,2, 3, 4, 6, 6, 7, «, and 9, with which the
^ater amiber must necessarily terminate.
■M
101 '.■• '-'t^l
■i -''i:
• (''■
POLITICAL ARITHMBTla
•■%
SINCE ^politicians liave acquired jvtstet id^as r^
sp^cting^hat constitutes the i^ stretigthr'of states,
vaHous researches have been made in r^rd to th&
number of the inhabitants in different ecNmfrieSj, itt
order to ascertain their |>opQlation. BesidesyaS-al-
nio^t all goyemmen^ Ime hett, under- the ii^desMty
of making loans, for the most peiH on anm^^^, tfiey;
hare naturall^r b^en induced to examine acctifdki^ td^
what progreission mankktd die, that the interest dftm to
loans ma;f be proportioned to the probability Of <&€»
aimulties becoming: eltinct. Thes^ calcidations hk^
been disdhguished by the ttame of PoRticat ArithmeH4 j
and, as it exhibits several cutiotis facts, whdbher coiii^
dered in a political or philosophicd point of vi^lif, #((
have thought it Our duty to give it a place h^r^, td
amuse and instruct our readers.
SECTION I.
Of the proportion betxoem thg males and the females.
Many people imagltie that the number of the females
born exceeds that of the males; but it has long sinc^
been proved, that the contrary is the case. More boys
than girls are bom every year; and since the y^
1631, a small interval excepted, we have a register tH
births in regard to sex ; and it has never been obser^i^,
that the number of the females bom ever even equaHed
that of the males. U is found, by taking a tbeati (A
average term in a great number of years, that the
?3
103 AMUSi^ENTS
number of the males born is to that of the females as
18 to 17. This proportion is nearly that which pre-
rails throughout all France; but,, to whatever reason
owing, it lieftnfl! at Paris ^ be as 527 to 26.
This kind of phenomenon is observed, not only in
England and France, but in every other country. We
may be convinced of the truth of it by inspecting the
calenders, which, at the commencement of every year,
give a table of the births that have taken place in most
of the capit^j xities of Europe ; it will be there seen,
Uiat the .nui|il](er of the males born always exceeds
t^at of the > females; and, consequently, it may be con-
sider/ad as a general law of nature.
Wie^m^.y: here observe a striking instance of the wis-
dom >Oi'^:^rqvidence, which has thus provided for the
presenra^pn of the human race. Men, in consequence
Qf th€ laptlve life for which they are naturally destined,
by their strength smd their courage, are exposed to more
dangers than the female sex; war, long, sea voyages^
€|9Cupa^ons laborious or prejudicial to health, and dis-
i^ips^tion, carry off great numbers of the males; and itr
th^^ results, that if the number bom of the latter did
not exceed, that of the females, the males would ra«
pidly decrease, and soon become extinct*
SECTION II.
Of the mortality of the human race, according to the Af-
ferent ages.
In this respect, there is apparently a considerable
difference between large towns and Uie country; but
this arises from the women in towns rarely suckling
their own children; and, consequently, the greater
part of their children being put out to nurse- in the
country, as it is in the period of childhood that the
greatest mortality prevails, it is most appturent in the
country. To make an exact calculation, it ought to he
founded on the deaths which happen in the towns,, as
well as in the country ; and this M. Dupr6 de St Maur
IN FOLITICiX ARnHUfETIC. 103
endeavoured to do, by comparing the registers of three
parishes in Paris, and-twelveinthe country.
According to the observations of this author, in
23994 deaths,- 6454 of them were those of children
not a year old. After carrying his researches on this
subject as far as possible, he concludes, that 24000
children bom, the numbers who attain to different ages,
are as follow :
Afet. Knraber.
2 years 17540
3 15162 --
4 14177
5 13477
6 12968
7 12562
8 12255
9 12015
10 11861
15 11405
20 . . 10909
25 10259
30 . 9544
35 . 8770
40 • 7929
45* 7008
5a 6197 ^
55 . • 5375
60 4564
65 . . 3450
70 2544
75 1507
80 807
85 291
90 .103
91 . . 71
92 . . .63
93 ... 47
94 40
95 33
96 • - . .23
V 4
101 jUTONsMiaiss 1 '
97 • • ♦ • •. • •. •.- • * •. r 18.
99 *k*^ »#«»,»• «• • Id
99 •••»»*•••♦.»• Q(
100 • • . . 6 Of 7.
^ Swchy then, is tbe eondhioii of the faiuii«D fpeeittf^
that, of 24000 children born, scarcely one half of tiiem
attain to the age of 9 ; and that two thirds are in their
grave before the age of 40. About a sixth only re-
main at the expiration of 62 years ; a ttntli after 70
years ; a hondredth part after 86 ; about a liiousandth
part attain to the age of 96 ; and six or se¥en indivi-
duals to that of 100.
By means of this table, we may ascertltn, pretty
nearly, what probability there is of a new4x)rn child
attaining lo a certain age; for this probability must be
to the contrary probabUity^ as the number of those who
attain to that age is to the number of those who die
before it.
For example^ as 4564 standi opposite ta 60, it in-
dicates, that as of 24000 children bom, there remain
no more than the above number of individnals at the
end of 60 'years^ 19436 must have died ; the proba-
bility, therefore, of a child attaining- to the tstge of 60,
is to the probability of its not attaiinng to ity as 4564
is to 19436. In this case, the ppoportion of those
living to those dead, fis nearly as 1 to 4; from which
we may conclude, that the chance • is 4 to 1 that a
new-born child will not attain to the age of 60.
If the probability of a person, of any determinate age,
living to another age,- be required ; for example, that
of a child of eight years of age attaining to the age of
60 ; we must compare the number of those who attain
to the age of 60^ with that of those who attain to the
age of 8 ; and the diflference, 7691, will give the num-
ber oi those who die between these two periods. We
shall then have this analogy: — a3^4564 is to 7691, so
is the probability that a- child of 8 years wttl attain to
the age of 60, to the probability of its not attainmg to
it. If 7691 be divided by 4564, it will be found that
the forttdf cobtaiAs the lattef nearly twicer and we
may therefore )say, that the ckcltice h almost 9 tiV 1,
that a diild of 8 years of age ^11 fiot li^e i» ilttct '
f 60,
SECTION Illi
Of the number of then of different ages in a given
number, '
t may be deduced from the pretieditig obsettatiOnSy
that when the inhabitants of a Cdutit^ amdtttit tO a
million, the number of those of the different ages will
be as follows :
Between and 1 year complete • 3^740
1 5» • • . . . .119460
5 10 . 99230
10 15 94530
15 20 , . 88673
20 25 ....*. . 82380""
25 30 77650
30 35 71665
35 40 64205-.
40 45 57230 j"
45 50 50605 \
50 bS 43940 \
bb 60 37110 I
60 65 28690
65 70 . -• 21305
70 75 . ,i . • . . . 13195
75 80 7065
80 85 2880
8^ 90 1025
90 95 ...... • 335
95 100 » 82
Above 100 years « 3 or 4.
Thus, in a cotmtry peopled with a millioa df inha- .
bitants, there are about 536350 between the age of 15
and 60 ; and, as neady one half of them are BSttn, e^
F 5
^•%-
106 .■■ . AJfUBEHfENTS
seqaaitly, this number of inhabitants could, on an
emer^encyy furnish 250 thousand men capable of bear*
ing armity even if an allowance be made for the sick,
lame, &c. who may be supposed to be among that
number.
SECTION IV.
Of the proportion of the births and deaths to the whole
number of the inhabitants of a country — The conse^
quences thence deduced.
As it would be difficult to number the inhabitants of
a country, and much more so to repeat the enumera-
tion as oflen as it might be necessary to ascertain the
population, means have been devised for accomplishing
the same object, by determming the proportion which,
the births and deaths bear to the whole number of the
inhabitants ; for, as registers of births and deaths are
regularly kept in all the civilized countries of Europe^
we may judge, by comparing them, whether the popu-
lation has increased or decreased : and, in the latter
case, can examine the causes which have produced the
diminution.
The proportion of the births to the whole population
in three generalities of France, which differ from each
other as much by the nature as the form of the soil,,
give the mean ratio of 1* to 25 J, without including the
large towns ; so that in this country we may reckon 51
inhabitants for two births.
But, as in towns of any magnitude, there are several
classes of citizens who spend their lives in celibacy,,
and who contribute eithemothing or very little to the
population, it is evident that the proportion between
the births and the effective inhabitants, must be more
considerable. It has been ascertained, by various
comparisons, that the proportion nearest the truth, is
that of 1 to 28 ; and it is this ratio which ought to be
employed^ in order to deduce from the births, in a
large city^the number of the inhabitants.
IN POLITICAL ARiniMETIC. 107
But there is reason to belieTe, that in regard to cities
of the first class, or capitab, such as London, Paris,
Amsterdam, &c. which are frequented by multitudes
of strangers, invited thither either by pleasure or busi-
ness, and where great luxury prevails, which increases
the number of those who live in voluntary celibacy, the
above proportion must be raised, and carried at least
to that of 1 to 30 or 31.
SECTION V.
Of some other proportions^ in regard to the inhabitants of'
a country.
We may deduce, by approximation, from the obser-
vations of various authors in England, France, Hol-
land, and Germany : —
1st. That the number of the inhabitants of a country
is to^thatof the families, as 1000 to 222 J ; so that
2000 inhabitants give, in general, 445 families.
2nd. That the number of the male children exceeds
that of the female ; and that this excess continue^ for.
more than 14 years, according to the proportion of
nearly 30 to 29. After 14 years, however, the number
of the females exceeds that of the males ^ in the propor-
tion of about 19 to 18, on account of the considerable
decrease of the males by war, navigation, laborious oc-
cupations, and intemperance.
3rd. That the number of the marriages is annually
to that of the inhabitants as 1 to 1 1 2.
4th. That the proportion of married men or widow-
ers, to the number of wives or widows, is nearly as 125
to 140; and the whole number of this class is to the
whole of the inhabitants as 53 to 126.
5 th. That the nimiber of widowers is to that of
widows nearly as 1 to 3. This at least is the propor-
tion deduced, from the enumeration of the people
made m Holland and in England. And it ought not
to appear astonishing, if it be considered, that most
men marry at a later period of life than the women,
and that Uieir laborious occupations, the maritime and
r 6
fatid imuir mivfaiek Ibef ttre engaged, md thcdhnCTsity
of the cHmfties which they fluent for th« take of
of commerce^ mast inertsase &o ntimber of the wido>w)i
iff the bilhr of mortality. ,
0th. That, admitting the above pi^oportion of widow^
enr and widows, it follows, that among' 651 inhabi*
tants, there are 118 married couples, firom 7 to d wi-
dowers, from 21 to 22 widows $ and the rest are com-
posed of children, persons in a state of celibacy, do-
mestics, and passengers.
7th. That 1870 married couples produce annually
357 diildren; for a towa having 10000 inhabitants
would contain that number of married couples, and
give annually 357 births ; from which it is concluded,
that 5 married couples, of all ages, give anniially, one
with another, one birth.
8 th. That the number of servants i^ to the whole
mxmber of inhabitants, nearly as 136 to 1535; which
IS a little more than the eleventh part. The number
of male domestics is nearly equal to that of the fomale;
being m the proportion of 67 to 69; but it is probable,
that in large cities, where great luxury prevails-, the
proportion must be difierent.
The above observations will enable us to solve the
following problem, and may serve to facilitate the solu-
tion of others relating to the same subject.
»
The age of a man being given^ mppose that of 30 years^
what probability is there that he xoiH be Mving at the
end of a determinate number of years, for example \5f
To resolve this problem, seek in the table of the
second section for the given age of the person, viz. SO,
and observe the number opposite to it, which w 9544 ;
then ford in the same table the number opposite to
45 X which is 700^^, and make the latter number the
numerator of a firaction, having for its denoxhiihttor the
former number. Th« fraction 1|J| will expreas the
probabiHty of a pcMoti of 30 attaining to the ag« of 45.
IN FOLnKaKL-AmSHMETIC. U39
The demoBstTBtisti of Ma rule wfll b& erkXent to
tlibse who muferstand the theory of probi^iUfies.
FROBL£M llr
Ayamg mm, aged %0, birnvm £1000. to he paidwkk
the mterest vtken he attmnf to the age of 25 ; but in
case he dies before that period, the debt to be cancelled t
xvkat sum omght the lender to receive at the proposed
term ofp^^ment T
- ^
It i» here evident, that if it were certain that the
young mati would live to complete his twentj-fiCth
year, the sum to he paid would he the capital increased -
with 5 years interest, which we shall suppose to be at
Ihe rate of 5 per cent, or £1250; But on account 'of
the lisk which the lender runs, by the chance of the
bom>wer dying before the time of payment, the sum
oo^t to be increased in the inverse ratio of the pro-
bability of his being alive. But this probability is ex-
pressed by the fraction tS^H'; and therefore the above
sum must be multiplied by thi» fraction inverted^ or
by T?4^, which will give nearly £1329. that is to say,
about 79 more for the risk of losing the money, which
certainly cannot be accounted usurious.
PROBLEM III.
A state, or ah individual, having occasion to borrow a sum
of money on an annuity, what interest ought to be given
for the different ages, the legal interest being 5 per
cent. ?
The vulgar, who are accustomed to burthensome
loans, entertain no doubt, that an annuity, at the rate
of 10 per cent, for the age of fifty, is a good bargain ;
and that this method of borrowing is advantageous to
the state, but they are egregiously mistaken; for it
appears .by the tsrbles of Pariceux, calculated from the
foregoing data, that 10 per cent, ought not to be given
before the age of 56. According to the same U.b\ft,^<^
110 AMUSEIUENTS
more than 6| per cent, ought to be given for the age
of 20 ; 6| for the age of 25; 6f for that of 36; 7f
for 40; 8 1 for 50; 10 at 56; ll^^atGO; I6f at 70;
274 at 80 ; and 39^ at 85.
It is therefore a great mistake to imagine, that on
account of the great number of persons who sink money
in these loans, on annuities, made by governments, the
latter are soon freed from paying a part of the annui-
ties by the death of a part of the annuitants. The slow
increase of annuities in tontines is a. sufficient proof of
the falsity of this idea ; besides, the greatness of the
number of the persons is precisely the cause, why the
extinction of the annuities takes place more in con-
formity to the laws of probability already explained. A
lucky chance, at the end of a few years, may free a
person from the payment of an annuity, established on
the life of a man 30 years of age; but if this annuity
were shared out on 300 different lives, the ages being
nearly the same, it is certain that he would not be libe-
rated from it before nearly 65 years ; and after 32 or
33 nearly one half of the annuitants would be living.
This Pariceux has clearly shewn, by examining tte
lists of the tontines.
HI
MAGIC SQUARES.
A MAGIC square is a series of figures arranged in
the cells of a square, in such a manner, that the figures
in each band, whether vertical, horizontal, or diagonal,
form exactly the same sum. They are divided into two
kinds : odd and even.
These squares have been called magic, because the
ancients ascribed to them great virtues, and because
this arrangement of numbers formed the basis and
principle of several of their talismans.
One square, containing unity, was, according to
them, the symbol of the Deity, on account of his unity
and immutability; for they observed, that this square,
by its nature, was single and immutable ; the product
of unity by itself being always unity.
A square containing four divisions or cells, was the
symbol of imperfect matter, on account of the impos*
sibility of arranging figures in it so as to form a magic
square.
The square, with 9 divisions, was consecrated to Sa-
turn; that with 16 to Jupiter; that with 25 to Mars;
that with 36 to the Sun ; that with 49 to Venus ; that
with 64 to Mercury; and that with 81 to the Moon.
Those who can find any relation between the ar-
rangement of numbers and the planets, must be indeed
not a little visibnary ; but such was the spirit of the
mysterious philosophy of lamblichus and Porphyry,
and of all their disciples, who were slaves to the most
stupid superstition, and to all the absurdities of judi-
cial astrology.
We shall here confine ourselves to the mechanical
method of forming a magical square^ either even or
odd.
113
AMUIXMEMT8
METHOD OF CONSTRUCTING AN ODD
SQUARE.
1st. Place unity below the middle celL
2nd. Place tbe followiiig numbea: in the cells which
descend diagonally from left to right.
3rd. When you come to the last diagonal cell, go
up to the -highest cell of the next following band.
4th. When the diagonal cell is filled up, carry the
next figure to the most distant cell on the left of the
lower band.
5th. In following* the diagonal, if rou meet with a
cell already filled up^^gpass over that ceff, and place the
figure in the 'diagonal from right to lefb. See the fol-
lowing figures, one of which represents a square of 9
divisions, and the^ other one of 25.
4
9
2
11
4
24
12
7
25
20
3
8
16
3
5
7
6
17
10
23
5
18
6
13
1
19
21
14
2
9
22
8
1
1
15
METHOD OF CONSTRUCTING AN EVEN
SQUARE.
We shaU apply this method to a square of 16 c^ls,
which is filled up in the following manner :
Ist^ Place 1 m the cell A (fig. M.) of the vertical
band <Mx the left; then pass the two next, and piace
4 in the u{^er edl of the perpendicular band, on the
right.
2nd. Omit 5, and place 6, 7, and the other figures,
as seen in fig. M#
The remaining 8 divisions, which are left vacamt,
must be filled up after the manner of fig. N. Reckfte
IN MMW CQViAES.
IIS
L ift tke <^ B witliout initftng i| dAwwif vbA ^lace 2
a&d 3 in the two next ceHs; their omit 4^ and set down
5 in the first cell of the next band ; omit 6 and 7, and
write down 8, and so on. If you then fill up each
of the&e squares from the other, you wiB haveiisqittoe
of 16 divisional — See the figures. . .
A OEOMEfTRiCAl/ SQUARE.
1
*
4
^
15
14
1
6
10
7
11
12
J
»
8
3
5
13
16
2
FIG. tf.
FIG. N.
To arrange in a square, consisting of 9 cells, the 9 terms of a
geometrical progression, in such a manner, that the pro*
duct arising Jrom the continued multiplication of the
numbers in each hand shaU he tihoays the same, and
equal to the cube of the middle term.
Let Ae terms of the progression be 1 : 2 : 4 : 8 :
16 : 32 : 64 : 128 : 256. Ifyou arrange these 9 terms
in a square of 9 cells, in the manner as you did the 9
terms of the arithmetical progression of the natural
numbers, 1,2, 3, &c. you will find that the product of
them, in every direction, amounts to 4096 ; which is_
exactly the cube of the middle term 16> as may be seen
in the annexed figure.
s
256
2
4
16
64
128
1
32
114
y.3:j(iifWEafSin« '^
To miAe'tke kmgkt pass overall the squares <^the chess
board, ome after the other, without passing twUe over the
same.
As the reader may peiiiaps be unacquainted with
the movement of the knight in the g^ame of cheiis, we
shall here describe it. If the knight be placed in the
square A, he. cannot be moved into any of the squares
in^nediately around him, as those marked 1^2, 3, 4,
5,. 6, 7, 8; nor into the squares 9, 10, 11, 12, which
are dir^tly above or . below, or on one side ; nor into
the squares 13, 14, 15, 16, which are in the diagonals^
hvk only into one of those which, in the figure,' are left
vai:ant. — See fig. B,
1
13i
10
14
■ • \
1
2
3
\ .
9
8
A
4
11
7
6
5
16
12
15
FIG. B.
Several celebrated men who amused themselves with
this problem, have given solutions of it ; but the fol-
lowing is the simplest of them all, and the easiest to be
remembered.
IN MAaie-.BfgaistEs.
34
49
23.
n
36
.
24
;
21
10
35
50
23
1-2
37
■It)
48,
33
64
57
38
26
2
13
9
ZO
51
54
63
60
53
41
14
26
32
47
68
61
56
3
19
8
55
52
59
62
27
42
46
31
6
17
44
99
A
15
7
18
45
30
16
43
28
The method consists in filling up, as much as possi-
ble, the exterior bands, which foim, as it were, a bor-
der, without entering the third, until there are no other
means of passing from the square at which you have
arrived to one of the two first ; a rule to which the
knight is necessarily subjected, in the most evident
manner, from his first step to the fiftieth. When he
arrives at the 50th, there is oo other choice than 51
or 63; but the 51st square being nearer the border,
ought to be preferred; and then his progress must
necessarilj be through 52, 53, 54, 55, 66, 67, 58, 59,
60, 61. When he arrives at the last, it is a matter
of indiSerence whether he be made to pass through the
three-Temaioing squiires, by directing his progress up-
wards or downwajds ; for in either case he will arrive
at the last
116 ikumKumm
APPLICATION OP ANALYSIS
TO THE
SOLUTION Olf VARIOUS PROBLEMS.
AS the object of tUs work k to unite instmction
with amusement, we shall confine ourselves to such
problems as are sufficiently easy to be sblfed by the
application of those rules which we have explained io
the introdite^ion. (14.)
PROBLEM I.
A lady lamenting that her age was triple that of her
daughter ; the latter eonsolea her by observing, that in
15 years it would be only double : what was the age of
eachf
Put a to denote the 15 years, and let x represent
the age of the daughter ; then by the conditions of the
problem, the ages of the daughter and mother, which
at present are x and 3 x, at the end of 15 years will
be or -f << sind 3 0? -f- a ; but as the age of the itiother
will then be double that of the daughter, we must
multiply the age of the latter by 2, to have the follow-
ing equation :
2.r-f2a=3x-f(i. (14.)
Then by transposition 2 a — a =: 3 x — 2 ,r, (15.)
And then by reduction a zz x.
IN AHMfm. 117
Consequently, the age of the daughter is 15, and
that of the mother 45; which will answer dl the
coiiditions of the problem.
FROBl'EM II,
A father, on his death-heiy gave orders in his wiU, that
if his wifcy who was then pregnant, brought fortii a
son, he should inherit j of his property, and the mother
the remainder ; but if she brought forth a daughter,
the latter should hoDe only i, and the mother ^ As
the widow, however, was delivered of tvnns, a hoy and
a girl, what share ought each to have of the property
l^'t by the father f
The only difficulty in this problem is to determine
what would hare been the will of ifte* testator, had he
foreseen that his wife would be delivered of twins. It
has generally been explained in the following manner:
As the testator desired that in case his wife brought
forth a son, he should have two thirds of his property,
and the mother one third, it hence follows, that his
intention was to give his son a sum double to that of
the mother; and as he desired, in the other case, that
if she brought forth a daughter, the mother should
have two thirds of his property, and the daughter one
third, there is reason to conclude, that he intended the
share of the mother to be double that of the daughter.
Ckmsequently, to unite these two conditions, the him-
•tage must be divided in such a manner, that the sqn
may have twice as much as the mother, and the mo-
ther twice as much as the daughter.
If o, therefore, be supposed to represent the father's
property, and ^ the share of the daughter, then £t will
express thajb of tbe mother^ and 4x that of the son.
But, as all these shares together are equal to the fa-
ther's property, we shall have the following equation :
By reduction lx:=. a
And by division a? = -. (16.)
118 ABttJinsMEinhs
Hence, if we suppose the whole property to be
£30000. the daughter's share will be <£4285f ; that
of the mother £85714, and that of tlie son £1714^«.
Sometimes the following difficulty is proposed in
regard to this problem. In case the mother should
be brought to bed of two sons and a daughter, in what
manner must the property be divided.
In our opinion, no other answet can be given than
what would be given by the gentlemen of the gown,
viz. that in this case the will would be void ; for as no
provision was made in it for a third child, its nullity
would be established according to all the laws hitherto
in existence. Because, 1st. Tiie law i& precise. 2nd.
Because it is impossible to determine what would have
been the dispositions of the testator if two sons had
been born to him, or if he . had foreseen that his wife
would be delivered of two.
PROBLEM III.
A captain being asked, how many soldiers he Aad in his
company, repliedrr-One half of them are in camp, one
third in the trenches, one eighth in the hospitaly and
four in prison. Of how mgjiy men did his company con-
sist ?
If the number of soldiers be expressed by x, and
the four in prison by c, we shall have the following
equation:
2 + 3-^ "8+^^^-
Then by mult. 24r + 16a? + 6a: 4- 48a = 4Sx.
By reduct. and transp. 48a = 48a? — 46x zz 2a:.
48a ^^
Consequently, x = — ^ = 9o.
IN ANALYSIS. 119
PROBLEM IT.
The head of a.Jish is 9 inches in lengthy its tail is as long
as the head and half the iody^ and this body is as lung
as the head and the tail. What is the length of the
Jish?
Let the head be expressed by a, the tail by Xy and
the body by y. By the conditions of the problem we
shall then have the following two equations :
y- a + fl^.^
(16.) By multipli- i 2jr = 2fl + y
cation. \ 2y :=: 2a -{• %a + y
By this result the problem is solved ; . for the head
being supposed equal to 9, the body dettoted by v=r
36, and the tail, being equal to the head and half the
body, must necessarily be^27, which answers all the
conditions of the problem.
PROBLEM V.
A person who had a lease of a house for 99 years j being
asked when it would exptre, replied, that two-thirds of
the time he had possessed it were exactly equal to four*
fifths of the time unexpired. How many years of the
lease were still remaining ?
If we call the time elapsed x, and the 99 years a,
the time unexpired will be a — x. Therefore, by the
conditions of the -problem,
2x 4a — 4j?
By multip. lOf =c 12a — 12«
Coosecj^ently, x :^ -r-- or -fr^ = 54.
1^ ilMWSII£Nn
Hence it appears, tkai as the time elapsed is 54
years, the period of the lease uoespired must neces-
sarily be 45; and this solution agrees with the condi-
tipns of the problem.
PROBI.SM VI.
his fr^poud to divide the nwnber 50 into tw6 such parts,
that the swn' ^ thre^'fetuihs ef the one, andjive^sivths
of the other may he equal fo 4Q.
Let 50 =: a, and 40 =: 6 ; and if one of the parts of
a be denoted by or, tiie other must necessarily be a .— x.
By the conditions of the (ffobl^m.WQ AhsU then have
the following equation :
3a: 6a — 6x
4 ^ 6
%^^^ \ 18:^-f 20« ^ «0x - tAh
„ J. . . 20fl~ 246
. By division ^- ^ — jczslOa — 126 = 20.
One of the parts of 50 dien is 20, and Xhe other 80 ;
which answers the conditions of the problem; for 15,
the three-fourths of 20, added to 25> the five-sixths of
50, ik justequaHo 40.
PROBLatfi vn.
- It is proposed to divide 100 into ivx) tuck partSy that if a
third of the one Be taken from a fourth of the other,,
the remainder shall be li«
Let 100 = a, and 11 =: 6; aho let one df the patts
_ j^
be expressed by x, and the other by a ~^x, Vken ~
will denote the tiurd^ the ^e part, and -^
IN ANALYSIS. 1?1
of the other ; and by the conditions of the problem we
shall have the following equation :
a — J? X -
By multipli. 3a — 3r — 4j?= \U
By transp } 3^ _ 12ft = 7;r
and redact. 3
By division — "^^ =: a? = 24.
The two parts of 100 then are 24- and 76; for if
8, the third of 24, be taken from 19, the fourth of 76,
the remainder will be II.
PROBLEM VIII.
Two persons sat down to play^ one of whom had 72 guineas,
and the other only 52 ; (ifter a certain nvmber '^' games
they separated, the former canying with him three times
as many guineas as the other. How much did he win ?
Let a represent the 72 guineas of the former, b the 52
guineas of the latter, and x the loss of the second
player.
The money of the first player when they give over
play will therefore be a + ^, and that of the other b—x ;
but as a -\- X, by the question, is three times as great
as 6 — X, we shall have :
a + X ^1 3b — 3x
By transp. 4j7 1= 3i — a
By division x = — "^ — =21.
4
As it here appears that the loss of the second player
was 21 guineas, leaving him only 31, the first must
have carried off 93 guineas, which answers the condi-
tions of the problem.
122 AMUSEMENTS
PROBLEM IX.
The minute hand of a clock being at \2^ and the hour hand
at I, at what point between 1 and 2 tciU they both be in
conjunction f
If X represent the space, between the hours of 1 and
2 passed over by the hour hand before it is orertaken
by the minute hand, and a the interval between 12 and
1 ; as the space passed over by the minute hand will
be twelve times as great as that passed over by the
hour hand, a -{- x will be equal to 12x; ^nd we sh^
have the following equation :
fl -}- Jf = 12x
By transp ) a-\\x
andreduct. j
By division y\^ ^*
From which we may conclude, that the minute hand
will overtake the hour hand after the latter has passed
over Vt P^^ of the space between the hours of \ smd 2.
PROBLEM X,
Jj two bodies move towards each other with unequal re/o-
citieSf the ratio of which is known, as well as the rfw-
tance between the bodies^ to determine the point at
which they will meet,
Let the velocities be as 12 to 1« and let a represent
the distance between the bodies, and x that part of it
passed over by the body having the least Telocity,
when they meet.
The space then passed over by the body which has
the greatest velocity, will he a^x, and we shall htre
the following proportion :
IN ANALYSIS. 123
12 : 1 :: a — X ; a?
By equation \2x zz a — x
By transp. I i3_ _ .
and reduct. 5 ^^^ - «
By division x = ^3.
The solution of this problem is general ; and conse-
quently applicable to all cases where the distance of the
bodies and the ratio of the velocities are known.
PROBLEM xn.
To divide 90 info two parts, which shall be to each ather
in the same ratio as 2 to 3.
«
Let 90 be represented by a, the least of the two
parts by x, and the other by a ^ x. We shall then
have the following proportion :
2 : 3 :: X : a ^ X
By equation 2a — . 2.r = 3x
By transp. ) Ofl - 5^
and reduct. j ^« - ^*
By division -— = r = 36.
Consequently, the least of the numbers will be 36,
and the other 54 ; and indeed 36 + 54 z= 90, and 36 :
54:: 2: 3.
PROBLEM XII.
Application of analysis to the solution of the Wth problem
of Divining Arithmetic, in which it is proposed to tell
the number of spots on all the bottom cards of several
heaps arranged on a table.
It is here supposed, that a complete pack of 52 cards
IS employed ; and that as many cards are placed ^s^t
62
f
134 AMUSEMENTS
the first of each heap as are necessary to make the sum
of the spots and cards together to amount to 12.
Let a represent 52, the whole number of cards, and
h that of the remaining cards. The number of cards in
all the heaps will then be a ^ ^. If the number of
spots to be guessed be expressed by x, smd the sum
of all these spots and the cards over them, as they are
known by c; we shall have the following equation:
X -}- fit — 6 :=c
By transp. a? = c -}- 6 — . a.
That is to say, if four heaps which are equivalent to
a be deducted I ^ will be equal to the sum of the re-
maining cards, and the number of the spots and cards
which are in the other heaps. The truth of this opera-
tion may be easily proved.
PROBLEM XIII.
What number is that, ^^^ y of i of 'which is equal to 1 ?
Let X be the required number.
Then * of ^^ = 1
3x 6x X
4 " 12^2
Consequently — = 1 , or a? =: 2.
But I of — = — = ;r
Proof: |of2are|; andJofi = |=l.
PROBLEM XIV.
jykat number is that i of ^ of which -f \ of \ of ity is
equal ^o 1 1 /
Let X, as before, be the required number.
Then j of | of j? are |
5x
And J of I of a: 13 ^
But by the supposition o" + T^ ^ ^^
Therefore 11jf= 11 x 12, and ar = 12.
IN ANALYSIS. 125
Froof: J of 12 are 8, and g of 8 are 6 = | of | of
12; I of 12 are 10, and the half of 10 is 5 = | of | of
12: but 5 + 6 = 11. Therefore, &c.
PROBLEM XV.
What number is that j of ^ of which — J of i of it are
equal to 19?
First, J of I of X is —
Sx
Aud 4'of 4 of X are rrz ;
* 15
Th S^ 3^— ^4ir^ 45x_ 19x
15 8 ""120 120"* 120*
19x
But — L equal 19 by the problem. Therefore, 19x =
19 X 120; and X equal 120.
Proof: |. of f of T^ry are 64, and J of | of 120 is
45 ; but 64 — 45 = 19. Therefore, &c.
PROBLEM XVI.
What number is that of which \of ^ multiplied hy\of^
of it will he equal to 6?
X
f of I of X are — ;
And I of iof X is ^, and | x ^ = ^^
Then by the conditions of the problem, — =: 6.
Therefore x* = 144 ; and consequently x = 12.
Troof: f of | of 12 are 6; and I of i of 12 is 1;
but 6 X 1 = 6. Therefore, &c
G 3
126 AMUSEMENTS
PROWLEM XVII.
fFhat number is that oftcTnch J + | are eqnalto iT
Let X be the number required.
__, X fix _ ox _
Therefore Sx rz 4
Consequently x iz f .
Proof: J of 4. = |; and | of | = 4 ; but | + | or
1^ 1= 1 . Therefore, &c.
PROBLEM XVIII.
What number is that the J, ^, and J o/" w/ficA mahe 12 f
Let J be the required number;
Then- + --fi-=12
2 3 4
Or 12jp -4- 8r + 6jc = 24 x 12'
Therefore 26.r = 288
And^=z\§^8-. ii_r_.
Proo/: i of lly-V is 5/^; 4. of llVy is 3^^; and J
of IItV is 2i|; but 5^ + S-^-j. + 2if =1 12.
PROBLEM XIK.
The triple f the half, and the fourth of a certain numhery
are equal to 104 : IFhat is the number 2
Let .r be the number required. We shall then have,
by the conditions of the problem :
3. +1+^ = 104
Therefore 30.r z= 1 04 >< 8 = 832
Consequently x = ?jV = ^"^tt-
In analysis. 137
Proof: 27\i x 3 = 83,^
J of 27-H- = 1344
4 of 27H = 6i|
The sum 104
PROBLEM XX.
tf I fln(/ ^ of the hull of a ship be immersed in the sea, and
only 4 feet of it above the surface of the water : What
is the depth of the vessel ?
Let X be the depth of the vessel.
r«i 3jc J? ^
Then -. + -4-4 = 0?
4 o
Or 18jp + 4a: 4- 96 = 24r
Therefore 2a? =: 96
And X =.48 feet, the depth of the vessel.
Proof: g of 48 = 36 . ;
J of 48= 8
44
Feet above water 4
48
PROBLEM XXI.
A banker at his death, being desirous to reward \0 of
his clerks, frave orders in his will, that 5500 guineas
should be divided among them, in such a manner, that
thefrst 5 should have each an equal share of the whole
legacy ; that the next 3 men should have shared among
them one-half of what was bequeathed to the first 5 ;
and that the 2 last should have divided between them
one-third of that sum : What was the share of each ?
Let X be the share of each of the first five clerks, and
a = the 5500 guineas.
138 A9IUSEM£14TS
Then, by the conditions of the problem, the share of
the first five will be 5x; that of the next three |jrv and
that of the two last -fr.
But as these three quantities are equal to a, or the
whole, we have the following equation:
5x + {x + l^r = a,
Bymultip. andreduct. 55x :^ 6a
By division a? r= — =: 600 guineas.
Eaich of the first five then had .... 600
Each of the next three 500
And each of the two last ......... .500
Froof: 5 x 600 = 3000
3 X 500 z= 1500
2 X 500 = 1000
5500
The application which we have here made of analysis
to the solution of a few problems, evidently shews that
this method, by its precision, brevity, and extent, is far
superior to arithmetic. The latter confines our atten-
tion to determinate quantities, and, if we may use the
expression, enchains it by the slowness of its progress;
"while the other, more rapid, enables us to pass over
the intermediate operations, and^to direct our attention
to the real point of difficulty.
The chief advantages, therefore, derived from this
science are, that it facilitates the discovery and com-
prehension of mathematical truths, and that it supplies
us with easy methods, and general rules, for resolving
all problems that may be proposed respecting quan-
tities.
When we have obtained a result by the rales of arith-
metic, thete is nothing indeed that exhibits to the mind
the chain of operations which conducted to it. When, af-
ter a few arithmetical operations, we have obtained 12 for
result, we see nothing in 12 which can indicate whether
this number has arisen from the multiplication of 3 by 4,
IN ANALYSIS. 1S9
of 2 by 6, or by the addition of 5 to 7, or of 21o 10 ; or,
in general, from the combination of any other opera-
tions. Arithmetic dves rules for finding certain results,
but these results of themselves can furnish no rules.-—
Algebra, however, or that mode of calculation which
employs indeterminate characters, preserves, as^we
may say, the traces of all the intermediate operations,
which conduct to the last result.
o 5
130 AMUSEMENTS.
TABLES OP CHANCES ON GAMES
OR PLAY.
The following tables contain the odds or chances^
for winning any number of games, in a great variiety
of cases, either when the chance is equal in every
game or throw, or when it is unequal according to
any odds or proportion; and the games may be of
any kind whatever, either at dice, or cards, or hazard^
or bilHardSy or racing, or cocking, &c.
I. When the chances or bet& on each game are equal.
Against winning
21 times running the odds are •• 2097151 to 1
20 out of 21, are 95324 to 1
19outof2l, are .*.... 9038 to 1
18 out of 21, are 1341 to 1
17 out of 21, are 276 tjo 1
16outof21, are 74 to 1
15 out of 21, are r*..^. 24} to 1
14 out of 21, •••« are 9} to 1
13outof21, are 4-J to 1
12 out of 21, ..-.- are ..►•^►•^ 2 to 1
20 game* running, ••••.. are •• 1048575 to 1
19 out of 20, ..• are .... 49931 to 1
18 out of 20, are 4968 to I
17 out of 20, are 775 to 1
16outof20,. ...• are • 168 to 1
15 out of 20, •• are ...--. 47^ to 1
14out€>f20, ...c....^.^. are Idltol
13outof20, •^•^ are 6j tol
12ouff20, are ^^tLr'''''Vf.^
' ( near* • o to 1
1 1 o«t of 20y •••^ are 4 to 3, or l^- to 1
even games in 20, are 48. 8d. to 1 s^. or 4|- to 1
TABLES OF CHANCES.
131
17 out of 19,
,16 out of 19.
ISout of 19,
14 out of 19,
l3out of 19,
12 out of 19,
11 out of 19,
1 8 games runr
17 out of 18,
16 out of 18,
l-'j out of 18,
14 out of 18,
13 out of 18,
12out of 18,
11 out of 18.
lOoutoflS, are^
....524287 to I
.••■ 26213 tol
2745 tol
450 tol
103 tol
30i to I
Snear 1 1 , or
lOAtol
4^101
5 2g. Id. to tB.
I orn(;ar2tol
..262143 tol
■ - 13796 tol
1523 tol
264 to]
.... 63 tol
.... I9Jtol
7^tol
-••• 3ftol
lear I0to7,or
nearer IfV to 1
17 gamea running ■
16outofl7,
15 out of 17,
'14 out of 17,
13 out of 17,
12 out of 17,
11 out of 17,
10 out of 17,
■ are -... 131071 tol
■ are 7280 to 1
■ are 850 to 1
■are 156 tol
■ are 39J
■ are 12-,^ or near 13 tol
o1
are 13tD6,ar2^tol
16 games running ■
ISoutofie,
14outofl6,
13oatofI6,
65535 tol
■ . 3854 to 1
.. 477 to]
•• 93 to]
132
AHUISEMISINTS.
12 out of 16 • . . . the odds are •
lloutof 16, are •
10 out of 16, are -
9outof.l6, arfe<
even games in 16, are
25 to
SJto
3|to
near? to 5; or
nearer i^to
4xr or near 4 to
] 5 games running
14 out of 15, U.
13 6utof 15, ..w
12 out of 15, •••
11 out of 15, •••
10 out of 15, ..•
^ out of 15, • • •
t • • •
• • •
are
are
are
are
are
are 5^ to
are23tolO,or2T'iyto
32767 to
2047 to
269 to
55 to
15Jto
... _ ( near 3|4, or
even games ml4.«.«.*«* are^ ♦•* ^
14 games running
13 out of 14, ...
12 out of 14, •••
11 out of 14, ...
10 out of 14, •••
9 out of 14, • . •
8 out of 14, •'••
are ••*» 16383 to
are 1091 to
are 153 to
are 33|, or near 34 to
are
are
near 3 to 2, or
IJto
are
{
10]- to
3|to
3^ to
8191 to
584 to
88 to
1 3 games runnmg ••.».• are
12 out of 1.3, •••! are
lloutof 13, are
I0outofl3, are -.v/j
9 out of 1 3, • • • are ej to
Cnear 9 to 4, or
«^4 2ito
204^ to
8 out of 13,
12 games running are 4095 to
11 out of 1^, .......... arfe .i.».. 314 to
10o\itofl5, .•.•*•••... are 50fJ or near 51 to
9 out of 1^, •••• are ..... 12|t6
TABLES OF CBANCE8.
ISS
•■i*«>Ai
8 out of 12,
7 out of 12,
• •••••••• are 4^, ornear4f tol
are^"^^'^ 8 to 5, olr
( . , near 1^ to 1
^.r^^ ^w^^^a ;« lo «»« $ ne2iX 34y, or near
eveti games in 12, •••••• are < « * ' ^^ *
11 games running,
10 out of 11, ..••
9 out of 11, ••••
8 out of 11, ••••
• •• ii »*
7 out of 11, •••••••••• are
are ....... 2047 tbl
are •••••• 1 69 to 1
are 29Jtol
are 7|^ or near 7^ to 1
' near 13 to 5, or
{
2^tol
10 games running
9 out of 10, •••<
8 out of 10, •••
7 out of 10, ....
6 out of 10, .*••
even games in 10,
1023 tol
9^ tol
174tol
53 to 1 1 , or near
44 tol
near 13 to 8, or
l^tol
are •••• near 3.j^ tol
are
are
are
are
are
{
tt>>^-
■•*i
9gamesrutming
8 out of 9, .^ • .
7 out of 9, ••••
( lO^iol
6outof9, i are 2||, or near 3 tol
• • »
• •• are 511 to 1
• •• are •••••• 50 tol
X 10^, or near
are
8 games running,'
7 out of B,
6otttof8, *•••<
5 out of 8, ••••
trm games in 8,
I
are •••••• 255 to 1
are a * * • • ^ • . 27J to 1
are 5f^, or near o t6 1
near 7 to 4, or
l|tol
C near 8 to 3, or
I 2|tpl
are
{
^e
mm
mm
mm
4S4 AUCSEMGNTS.
. . 127 to 1
.... 15 tol
r 17 to 5, or
3^ tol
S games running;, ••■•■■ are 63 to 1
5 out of 6, ■•• ' are 8^ t '
A. «,.. ^« C21 to 11, or
loutoffi, •••■ are J ' „
in 6, are 1 1 to 5, or 2 ■ to 1
5 gamcE Tunning, are 31 tol
■~ o3,or
44 tol
4 out of 5,
4 games running, •...-■ are •••■•••■ 15 tol
3 out of 4, are 1 1 to 5, or ii to
games in 4, are --StoS.or l|to
jl3to3,t
3 games running, ■ ■
2 games running, ••
II, When ihe Odds or Chances on each game.
Against winning
10 games running, the odds are 427 tol
9ontofl0, are 44 tol
8outoflO, are 9Jtol
7 out of 10, are 2^,ornear3 tol
eontoflO. ^ ( nearly equal, or
t iT^Tto'
even games ml are 3^ tol
games running, are 232 tol
8outof9, are 261 tol
7 out of 9, are very near € tol
6outof9, £„4nearl5to8
5 out of 9, •• are ■
near 14 to 9, or
14 tol
TiBLES OF CHAVCG3.
iSS
games runnings
out of 8,
out of 8,
6 out of 8,
games in 8, •
7 games runnipg .
" tor?,
t of?
4ont of 7,
games running .
S out of 6,
4 out of 6
5 games n
4 out of 5.
4 games run
3 out of 4,
3 games running,
2 out of 3,
126 1
■ I5itol
e ioto4, orSjtol
e lltolO.orlVCTtol
e 14 to 5, or 2^ to 1
c 68 tol
e 94*0!
■e 9 to 4. or 2J to 1
e near3to2orlitol
e 3fi:
■37 to I
SJtol
3tolO,orl^tol
.9to4,or2itol
195.1.
rather better
than 3 tc
near 7 to 5, or
Ut
10|tt)l
near 8 to 5, or
litol
near 6 to 3,
l|tal
near 31 to 6,
5^tol
near 10 to 7,
2 games runnmg,
I game each, out of 2, .
t near 7 to 3, or
/near 61 to 60, w
«156 4MUIE1IENT8.
SI. Whtn. Ike Odd* or Chtmca on each
Agunfit winning
10 games running are . . .
game are &toG.
.. 2654 tol
203 to 1
.... 33 to 1
8^ to 1
to 7, or 24 tol
are
7outof]0, .....
9 games running.
. . . 1206 to 1
.... 101 to I
.... njtoi
4Jtol
are
are
8 games running.
are ...
... 647 tol
... SOJtol
9ltol
ar 19 to 7, or
2^101
are
-5 out of 8,
-r
7 games running .
8 out of 7, ....
are ...
.... 248 to 1
.... 25itol
5JtoI
6 gamcB running.
... 112 tol
... 123 tol
14 to 5, or
24 tol
4 out of 6,
... are|""
£ games running,
.... SOJto]
... C^tol
...
4 game* running,
22^ tol
ynearly3B'TWl
'
are ver
3 games running,
2 games running,
are •■•
are ...
. nearSJtol
. near3|tol
TABLES OF CHAKCES.
137
IV. When the Oddi or Chanu
Against winning
10 games rannuig
9 out of 10, • >
8 out of 10, • •
7 out of 10, . .
6 out of 10, • •
even games in 10,
9 games running, •
e out of 9, ■
7 out of 9, .
5 out of 9, •
5 out of 9, ■
8 games running, •
7 out of 8, .
6 out of 8,' .
5 out of 8, .
even games in
7 games running ■
6 out of 7, . .
4 out of 7, ■
6 games ninnmg •
6 out of 6, •
4 out of 6, •
even games in I
5 games running '
4 out of 6, . '
3 out of 5, .
im each game are 5 to 4.
356 tor
38jtol
fi^tol
21toi
- 14tol3,orlV^tol
• • 13to4,or34tol
197 to 1
24 tol
5JtoI
Qe&rftoS, or If' tol
• 12to7,orl|tol
109 tol
I3itoi
■ • • . • 34 tol
T22t«2I,orl^tol
ear 17 to 6, or 2i to 1
nearS to3, or 11
) tol
Sitol
33 tol
near39to8,or4|tol
near 6 to 5, or 1|to ]
near 11 to 6,or2-Jtol
17Jtol
near 14to5, orZ^to 1
near3to'2, orl^tol
. very ne!ir 9| to
near 3 to 2, or I ^ to I
jarl7 tolO.orlVs
138
AMUSEMSMTfi
3 games running, •
2 Out of 3, . • .
. are i
* are
near 4|, or near 5 to I
near 4 to 3, or l|to I
2 games running • • are
] game each out of 2, are
• _.L_
56 to 25, or near 2^ to 1
4lto40, or l^tol
V. When the Odds or i
Against winning
10 games running, •
9 out of 10, . . •
8 out of 10, . V •
7 out of 10, • • •
6 out of 10, • . •
Chances
• are '
• are
. are
• are •
• are <
on each game are 4 to 5.
. . . • • 3324 tol
245 tol
38^ to]
. ..... 9|tol
3 to 1
9 games running, •
8 out of 9, • • •
7 out of 9, • • •
6 out of 9, • • •
• are
• are *
• are <
• are
1476 tol
119 tol
, . . . . 20*. tol
5|tol
8 games running, •
7 out of 8, • • •
6 out of 8, ...
5 out of 8, • • •
. are <
• are -
• are
• are •
655 tol
58 to 1
. ^ . . . near 11 to 1
3 tol
7 games running •
6 out of 7, ...
5 out of 7, . I •
. are •
• are
• are *
290 to I
28|tol
• • • 5|, or near 6 to 1
6 games running •
5 out of 6, • •^ •
4 out of 6, ...
• are .
• are •
. are •
.... 128 to J
• • • • near 142 tol
» . . . . 3 tol
6 games running .
4 out of 5, ...
. are <
• are •
56i to 1
» . near6fj, or7 tol
4 games running •
3 out of 4, • • •
• are
• are
24J to 1
3 to 1
3 games running, •
2 games running, •
• are
• are <
• 1 ^It* or near 1 0^- to 1
. . 65tol6,or4T3^tol
tABLES 6V enkHC&i.
139
VI. When the Odds or Chances on each game are
6 to 4.
Against winning
6 times running
5 out of 69
• •
4 out of 6, •
even games in 6
are # . . . near204tol
are * • • • near 3f>lo 1
J 1601 to 14^4, Of
( near l^to 1
are • • • • • near 2|- to 1
are
5 times runtiing,
4 out of 5, • •
3 out of 5, • •
are • • • • near ll^to^l
are • • • very near 2 toT
are near 31 to 15, or 2^^^ 1
4 times running
3 out of 4, • •
even games in 4,
are* • • • • near6|^tol
are 328 to 297, or II to 10
are • 409 to 216, or 15 to 8
3 time» running
2 Out of 3, • «
are . • 3^^, ornear^tol
are 81 to 44, or near 1^ to 1
2 times running, • • are • • 16 to 9, or 1^ to 1
1 game only in 2 • • are • • 13 to 12, or 1-^ to 1
VII. JVhen the Chances on each game are 4 fo 6»
Against winning
& times running • • are
5 out of 6, . . • . are • • •
4 out of 6, .... are near 32 to 7, or 4^ to 1
• • • .
.243 tol
near 23| to 1
6 times running, • • are ...... 96 to 1
4 out of 5, .... are .... near I0| to 1
4 times running, • . are 38-,^ to 1
3 out of 4 .... are .••• • near 4 J to ]
140
3 limes ruDning, ■
2 times running, .
Zl
l*|toi
. 21 to4,or5itol
VII. U'hfn the Oddi
Against winning
6 times running ■
5 out of 6, ■ . •
4 out of 6, • . ■
equal times in 6 . ■
on each game are 7 to 4.
are l+^tol
are . nearl2 to 5,or2^tol
are nearl09to67,or5to3
are ■ • • very near 3 to 1
fi times running, •
4 out of 5, ■ ■ •
3 out of £, • • •
• are
. are
near 8^ to 1
near 97 to 65, or 3 to 2
■(near 29 to 10, or
J near 3 to 1
4 times running ■
3 out of 4, ■ . .
. are
fil tolO.ornearStol
.n«;3rl9to9,or2itol
3 times nurning ■
2 out of 3, ...
.are
( near 23 to 8, or
I near 3 tol
. . . 7 to 3, or 2| to I
2 times running ■
lin2,c,reveuiQ2,
".r
. .72to49,orlTVto]
. ■ 65to56,or7to6
IX. /FAfn tht Chance on each game is 4 to 7.
Against winning
5 limes running, ■ - are 431 to!
5 out of 6, . . ■ ■ are 36J to I
4 out of 6, .... are 6i to 1
6 times running, ■
, 4outof5, . . .
'.Z
156 tol
e ]5{.toI
TABLES OF CHANCES. 141
4 times running, • • are 56 to" 1
3 out of 4, . . • . are . . • • • 6^. to 1
3 times running, • • af& 19^ to 1
2 times running, • • are • • • • • 6/^tol
X. When the Odds on each game are 2 to I,
Against winning
6 times running, • • are • • • near lOytol
5 out of 6, • . • . are 473 to 256, or 1 1 to 6
4outof6, • • • -are near 17 to 8, or 2-}. to I
3 out of 6, or even • are • near 7 to 2, or 3| to 1
5 times running, • • are near 33 to 5, or 6-|- to 1
4 out of 5, • • • • are 131 to 1 12, or near 7 166
3 out of 5, • • • • are • near 34 to 9, or 3 J to I
4 times running, . • are 65 to 1 6, or near 4 to 1
3outor4, • . . • are • l6to 11, or l-j5j. tol
2 out of 4, or even in 4, are • • • 1 9 to 8, or 2\ to 1
3 times running, • • are* • • • 19 to 7, or 2^ to 1
2 out of 3, . . • • are • 20 to 7, or near 3 to 1
2 times running, • • are • • • 5 to 4, or 1 ^ to 1
1 in 2, or even in 2, • are • • • 5 to 4, or l| to 1
XI. When the Chance on each game is J, or 1 to 2.
Against winning
6 times running, • • are • • • • • 728 to 1
5 out of 6, • • • • are 55 to 1
4 out of 6, . . . • are • • 8^, or near 9 to 1
142
AMUSEMENTS.
S times ninnlDg, • « are • • • • * 24^ to 1
4 out of 5, • • • • are • 21-pi-, or near 21 to l
4 times runmng, • • are • • • • • 60 to 1
3 out of 4, • • • • are 8 to 1
3 times running, • • are . * . • . 26 to 1
2 times running, • • are 8 to 1
■ ra I t. ffniMf-'nr •»■> i I ir t mi nt m " ^•'-
i' ^
143
ACOUSTICS AND MUSIC.
THE ancients seem to have considered sounds under
no other point of view than that of music ; that is to
say, as affecting the ear in an agreeable manner : it is
even very doubtful whether they were acquainted with
any thing more than melody, and whether they had
any art similar to that which we call composition.
The moderns, however, by attending to the philosophy
of sounds, have made many discoveries in this depart-
ment, so much neglected by the ancients; and hence
has arisen a new science, distinguished by the name of
Acoustics. Acoustics have for their object the nature
of sounds considered, in general, both in a mathema-
tical and philosophical view. This science, therefore,
comprehends ipusic, which considers the ratios of
sounds, so far as they are agreeable to the ear, either
by their succession, which constitutes melody, or by
their simultaneiety, which forms harmony. We shall
here give a brief account of every thing most curious
and interesting in regard to this science.
Defittitian of sound ; how diffused and transmitted to our
organs qf hearing — experiments on this svbject'-^if-
ferent imays of producing sound,
Sound is nothing else but the vibration of the parti-
cles of the air, Qccasioijied either by some sudden agi-
tation of a certain mass of the atmosphere, violently
compressed or exp^ded, or by the communication of
the vibration of the insensible parts of a bard and elas-
tic body.
l44
AMUSEMENTS.
These are the two best known ways of producing
sound. The explosion of a pistol, or of any other kind
of fire-arms, produces a report or sound, because the
air or elastic fluid, contained in the gunpowder, being
suddenly dilated, compresses the external air with
great violence : the latter, in consequence of its elas-
ticity, re-acts on the surrounding atmosphere, and
produces in its moleculee an oscillatory motion, which
occasions the sound, and which extends to a greater
or less distance, according to .the intensity of the cause
that gave rise to it.
The other method of producing sound, is to excite
in an elastic body, vibrations sufficiently rapid to oc-
casion, in the surrounding parts of the air, a similar
motion. Thus an extended string, when struck, emits
a sound ; and its oscillations, that is to say, its motion
backward and forward, may be distinctly seen. The
elastic parts of the air struck by the string, during the
time it is vibrating, are themselves put into a state of
vibration, and communicate this motion to the neigh-
bouring ones. Such also is the mechanism by which
a bell produces its sound : when struck, its vibrations
are sensible to the hand which touches it.
That air is the vehicle of sound, may be proved by
the following experiment: if a bell be suspended in
the receiver of an air-pump, the sound of it decreases
in proportion as the air is exhausted, and at last be-
comes totally insensible when a complete vacuum has
been formed.
Sound always ceases when the vibrations of t^ie so-
norous body oease, or become too weak. This may
be proved also by an experiment; for when the vibra-
tions of a sonorous body are damped by any soft body,
the sound seems suddenly to cease ; in a piano-forte,
therefore, the quills are furnished with bits of cloth,
that by touching the strings when they fall down, they
may damp their vibrations. On the other hand, when
the sonorous body is, by its nature, capable of conti-
nuing its vibrations for a considerable time, as is the
case with a large bell, the sound may be hesgrd for a
long time after.
jl
IN ACOUSTICS, 145
Of the velocity of sound; experiments for determining it —
method of measuring distances hy it.
Light is transmitted from one place to another with
inconceivable velocity; but this is not the case witfi
sound : the velocity of sound is very moderate, and
may be measured in the following manner.
Let a cannon be placed at the distance of several
thousand yards, and let an observer, with a pendulum
that vibrates seconds, or rather half-seconds, put the
pendulum in motion, as soon as he sees the flash, and
then count the number of seconds or half-seconds
which elapse between that period and the moment when
he hears the explosion. It is evident, that if the mo-
ment when the flash is seen be considered as the sig-
nal of the explosion, nothing will be necessary to ob-
tain the number of yards which the sound has passed
over in a second or half-second^ but to divide the
number of the yards, between the place of observation
and the cannon, by the number of the seconds or half-
seconds which have been counted.
Now the moment when the flash is perceived may be
considered as the real moment of the explosion ; for
80 great is the velocity of light, that it employs scarcely
a second to traverse 70000 leagues.
By this method it has been found, that sound moves
at the rate of about 1142 feet in a second.
This method may be employed to determine the dis-
tance of ships at sea, or in a harbour, when they fire
guns, provided the flash can be seen, and the explosion
heard. During a storm also the distance of a thunder-
cloud may be determined in the same manner. But,
as a pendulum is not always to be obtained, its place
may be supplied by observing the beats of the pulse ;
for when in its usual state, each interval between the
pultatiom is almost jequal to a second.
K
146 ^ AMUS^ENtS
It(m sounds may ht pr(fpagated in every direction^ without
confusion.
This is a very singular phenomenon in the propaga-
tion of sounds; for if several persons speak at the
same time, or play on instruments, their different
sounds are heard simultaneously, or.all together, either
by one person, or by several persons, without being
confounded in passing through the same place in dif-
ferent directions. Let us endeavour to account for
this phenomenon.
The moleculee of the air contiguous to the sonorous
body, receive from it an oscillatory and vibratory mo-
tion, which, inconsequence of .their elasticity, is suc-
cessively transmitted to a certain distsmce. ^s the
sonorous body is the centre from which the motion is
communicated in every direction, the sound must ne-
cessarily become weaker in proportion as the mass of
air, which receives it, becomes greater. The differcBt
sounds, of whatever nature, must be heard, beoailBe
they are tiansmitted to the organ of hearing by analo-
gous moleculse of the air, in the same manner as when
a certain tone is emitted, in an apartment, it cannot
be repeated but by those strings of the instrument
which are in unison with it. Sounds of greater inten-
sity cannot be propagated with more velocity, though
the vibrations of the serial moleculee which transmit
them be stronger, because they are always isochronous,
like those of pendulums more or less removed from th«
vertical line, or strings more or less bent.
Of echoes; — how produced — account of the most remark*
able echoes y and of some phenomena respecting them.
Echoes are well known ; but however common this
phenomenon may be, it must be allowed that the man-
ner in which it is produced, is involved in considerable
obscurity ; and that the explanation given of it doe«
not sufficiently account for all the circumstances at-
tending it.
IN ACOUSTICS. 147
All philosophers almost have ascribed the formation
of echoes to a reflection of sound, similar to that ex-
perienced by light, when it falls on a polished body;
but, as D'Alembert observes, this explanation is false ;
if it were not, a polished surface would be necessary
for the production of an echo ; —but it is well known
that this is not the case. Echoes indeed are frequently
heard opposite to old walls, which are far from beimg
polished; near shapeless masses of rock, and in the
neighbourhood of forests, and even of clouds. This
reflection of sound, therefore, is not of the same nature
as that of light.
It is evident, however, that the formation of an echo
ean be ascribed only to the repercussion of sound;
for echoes are never heard but when sound is inter-
cepted and made to rebound by one or more obsta-
cles.
Sound, as already said, is propagated in every direc-
tion by the vibration of the particles of the air ; but
if any column of air rests against some obstacle that
prevents the direct movement of the elastic globules,
which serve as the vehicle of sound, it must rebound
in a contrary direction, and striking the ear, if it meets
with one in the line of repercussion, convey to it a re-
petition of the same sound, provided the original
' sound does not affect that organ at the same instant.
But we are taught by experience, that the ear does
not distinguish the succession of two sounds, unless
there be between them tlie interval of at least one
twelfth of a second; for during the most rapid move-
ment of instrumental music, each measure of which
cannot be estimated at less than a second, twelve not«s
are the utmost that can be comprehended in a mea-
sure, to render the succession of the sounds distin-
guishable; consequently the obstacle, which reflects
the sound, must be at such a distance, that the rever-
berated sound shall not succeed the direct sound, till
after one twelfth of a second; and as sound moves at
the tate ef about 1142 feet in a secbnd, and conse-
.qHQQtly about 95 feet in the twelfth of a second, it
thence follows, that to render the rev%\be.tAX.^^ ^Q»>xt^
u2
148 AMUSEMENTSr
distinguishable from the direct sound, the obstacle
must be at the distance of no more than about 48
feet ■"*'"
There are single and compound echoes. In the for-
mer, only one repetition of the sound is heard; in the
latter, there are 2, 3, 4, 5, &c. repetitions. We are
even told of echoes that can repeat the same void 40
or 60 times.
Single echoes are those where there is only one ob-
stacle ; but double, triple, or quadruple echoes, give
us reason to suppose several obstacles disposed in such
a manner, that the diflTerent reflected sounds strike the
ear at times sensibly different.
There are some echoes that repeat several words in
succession ; but this is not astonishing, and must al-
ways be the case when a person is at such a distance
from the echo, that there is sufficiient time to pro-
nounce several words before the repetition of the first
has reached the ear.
There are certain echoes which have been much ce-
lebrated on account of their singularity, or of the num-
ber of times that they repeat the same word. Misson,
in his description of Italy, speaks of an echo, in the
vineyard of Simonetta, which repeated the same word
40 times.
At Woodstock, in Oxfordshire, there is an echo
which repeats the same sound 50 times.
The description of an echo still more singular, near
Rosneath, some miles distant from Glasgow, ipay be
found in the Philosophical Transactions for the year
1698. If a person, placed at the proper distance,
plays 8 or 10 notes of an air with a trumpet, the echo
faithfully repeats them, but a third lower; after a short
silence, another repetition is heard, in a tone still
low6r; and another short silence is followed by a
tliird repetition, in a tone a third lower.
A similar phenomenon observed in some places is,
that if a person stands in a certain position, and pro-
nounces a few words with a low voice, they are heard
only by another person standing in another determi-
nate place : this arises from the elliptic form of arches,
IN ACOUSTICS. 149
which have the property of collecting in one of thei^
foci the rays that proceed diverging from the other.
The following phenomenon depends on the same
theory.
To construct two figures, to he plaped at the two ends
of a kally one of which shall repeat to the ear of a per*
son what has been whispered into the ear of the other
Jigure, without being heard by any other person in the
hall.
Provide two heads or busts, made of pasteboard^
resting on pedestals, and place them in a hall at such
a distance from each other as you may think proper*
Then convey a tube of tin-plate, an inch in diameter,
from the ear of one of the figures, through the pedes*
tal on which it rests, and below the flooring, till it
reach the mouth of the other figure, passing through
its pedestal in the same manner as that of the former;,
this tube must be a little wider, at each of its extre-
mities, somewhat in the form of a funnel.
When it is necessary to bend this tube, care must be
taken to cover the interior angles with a piece of tin-
plate inclined at an angle of 45 degrees, that the voice
may be directly reflected from one part of the tube to
the other, and that the sound may be conveyed dis-
tinctly to the ear.
This construction will produce the following effect.
If a person whispers into the ear of one of these figures,
the words he pronounces will be distinctly heard by a
second person who applies his ear to the mouth of the
other figure.
The secret of the magic mirror, as it is called, de-
pends on the same theory. The construction of this
muTor is as follows:
Fix, in a vertical position, a concave mirror, two feet
in diameter, and of such a degree of curviture, that
the focus of the rays which fall upon it, in a parallel
direction, may be at the distance of twelve or fifteen
inches from the reflecting surface. At this distance
' H 3
150 AMUSEMENTS
place a small figure, but in such a mannefi that iti
head may be exactly in the focus.
This mirror must be placed at the distance of eight
or ten feet from a wall opposite to it, and parallel to
its surface : the wall must have in it an aperture,- equal
to the surface of the mirror, concealed by a very fine
curtain, that the sound may easily pabs through it.
Provide also a second mirror of the same form, with a
similai? figure, and place it behind the wall at the dis-
tance of two or three feet from it, and opposite to the
former, with the figure in its focus. It may be readily
conceived, that when a person only whispers into the
ear of the small figure behind the wall, a person stand-
ing near that placed in the focus of the opposite mir-
ror, will hear very distinctly the words whispered into
the ear of the former. In this manner, the person
\tho asks a question, standing near the first figure,
hears the answer which is whispered into the ear of the
Other behind the wall.
In order to conceal entirely the apparatus which pro-
duces this effect, and to render it much more extraor-
dinary, the pretended concave magic mirror may be
covered with a piece of gauze, which will not prevent
the transmission of the sounds firom the one focus to
the other.
The Memoirs of the Academy of Sciences at Paris,
for the year 1692, speak of a very remarkable echo in
the court of a gentleman's seat, called Le Genetay, in
the neighbourhood of Rouen. It is attended with this
singular phenomenon, that a person who sings or speaks
in a low tone does not hear the repetition of the echo,
but only his own voice ; while, on the other hand, those
who listen hear only the repetition of the echo, but
with surprising variations ; for the echo seems some-
times to approach and sometimes to recede, and at
length ceases when the person who speaks removes to
some distance in a certain direction. Sometimes only
one voice is heard, sometimes several, and sometimes
one is heard in the right, and another on the left. An
explanation of all these phenomena, deduced from the
semi-circular form of the court, may be seen in the
above collection.
ON MUSICAL STBINGS. 1£|1
EXPERIMENTS RESPECTING THE VIBRATIONS
OF MUSICAL STRINGS, WHICH FORM THE
BASIS OF THE THEORY OF MUSIC.
If a string of metal or cat-gut, such as is used for
musical instruments, made fast at one of its extremi-
ties, be extended in a horizontal direction over a fixed
bridge, and a weight be suspended from the other ex-
tremity, so as to stretch it ; this string, when struck,
will emit a sound produced by reciprocal vibrations
which are sensible to the sight.
If the part of the string made to vibrate be shortened,
and reduced to one half of its length, any person
who has a musical ear will observe, that the new sound
is the octave of the former : that is to say, twice as
sharp.
If the vibrating part of the string be reduced to two-
thirds of the original length, the sound it emits will be
the fifth of the first.
If the length be reduced to three-fourths, it will give
the fourth of the first.
If it be reduced to ^, it will give the third major j
if to |., the third minor. If reduced to |^, it will give
what i% called the tone major; if to t^, the ton^
minor; and if to 4|» the semi-tone, or that which in
the gamut is between mi and /*«, or «and soL
The same results will be obtained if a string be fas-
tened at both ends, and ^, y, and | of it be succes-
sively intercepted by means of a moveable bridge.
(SeethcfoUomng table relating to this subject,) •
uA
AMUSEMENTS
ON MUSICAL STRINGS. 153
Such is the result of a determinate degree -ortension
applied to a string, when the length of it has been made
lo vary. Let us now suppose that the length of the
string is constantly the same, but that its degree of
tension is varied. The following is what we are taught
by experiment on this subject:
If a weight be suspended at one end of a string of
a determinate length, made fast by the other, and if
the ton5 it emits be fixed, when another weight qua-
druple of the former is applied, the tone will be the
octave of the former ; if the v^reight be nine times as
heavy, the tone will be the octave of the fifth ; and so
on : so that the tones will become acute in the ratio
of the square roots of the weights.
The size of the strings has an effect in regard to the
tones, as well as the different lengths of the string,
and the weight by which it is stretched ; for it is proved
by experiment, that a string twice as small in diameter
as another, ♦every thing else being the same, emits a
tcne which is the octave of that of the other; and that
if the diameter is only a third of that of the other, the
tone is the octave of the fifth of that other string, fol-
lowing the order of the diatonic scale.
We may thence conclude, that the tones of the mu-
sical strings are in the direct ratio of the square root of
the weights by which they are stretched, and in the
inverse ratio of the lengths and diameters of these
strings.
Consequently, to bring into unison strings which
differ in length and diameter, and which are stretched
by different weights, the compound ratio thence result-
ing must be exactly the same, in order that the fre-
quency of the vibration in one may be compensated by
the slowness of another. Thus, two strings of the
same size, the lengths of which are as 2 to 1, and the
stretching weights as 4 to 1 , will have their vibrations
isochronous, that is to say, they will be in unison.:
two strings, the diameters of which are as 2 to 1, and
the lengths as 1 to 2, stretched by equal weights, will
be in unison also, as well as those which; being of
u 5
154 AMUSEMENTg
equal lengths, have their diameters as 2 to 1, and the
stretching weight as 4 to 1 .
We may conclude, therefore, that two strings, the
•diameters of which are as 3 to 2, and the lengths as
1 to 3, cannot be in unison, unless the weights, by
which they are stretched, be to each other in the same
ratio as 1 to 4.
To determine the number of the vibrations made hy a stiing
of' a given length and size, when stretched bif a given
%veight.
A very ingenious method, invented by M. Sauyeur,
for finding the number of these vibrations, may be seeu
in the Memoirs of the Academy of Sciences, for 1700.
Having observed, when two organ-pipes, very low,
and having tones very near to each other, were sounded
at the same time, that a series of pulsations or beat»
were heard in the sounds ; and by reflecting on the
cause of this phenomenon, he found that these beats
arose from the periodical meeting of the coincident
vibrations of the two pipes. Hence he concluded, that
if^the number of these pulsations, which took place in a
second, could be ascertained by a stop watch, and if it
were possible also to determine, by the nature of the
consonance of the two pipes, the ratio of the vibra-
tions which they made in the same time, he should be
able to ascertain the real number of the vibrations
made by each.
. We shall here suppose, for example, that two organ*
pipes are exactly tuned, the one to mi flat, and the
other to mi : it is well known, that as the interval be-
tween these two tones is a semi-tone minor, expressed
by the ratio of 24 to 25, the higher pipe will perform
25 vibrations while the lower performs only 24; so
that at each 25th vibration of the former, or the 24th
of the latter, there will be a pulsation : if 6 pulsations,
therefore are observed in the course of 1 second, we
ought to conclude, that 24 vibrations of the one and
25 of the other take place in the tentlr of a second :
ON MUSICAL STRINGS. 155
ftnd consequently, that the one performs 240 vibrations,
and the other 250, in the course of a second,
M. Sauveur made experiments according to this idea,
and found that an open organ-pipe, 5 feet in length,
make)B 100 vibrations per second; consequently, one of
4 feet, which gives the lower triple octave, and the
lowest sound perceptible to the ear, would make only
12 J ; on the other hand, a pipe of one inch less t^,
being the shortest the sound of which can be distin-
guished, will give in a second 6400 vibrations. The
fimits, therefore, of the slowest and the quickest vibra-
tions appreciable by the ear, are, according to M. Sau-
veur, 12i and 6400.
We shall not enlarge further on these details, but
proceed to a very curious phenomenon respecting strings
in a state of vibration.
Make fast a string by both its extremities, and by
means of a bridge divide it into aliquot parts, for ex-
ample, 3 on the one side, and 1 on the other, and pat
the larger part, that is to say, the j, in a state of vibra*
tion; if the bridge absolutely intercepts all communi-
cation from the one part to the other, these | of the
string, as is well known, will give the tone of the fourth
of the whole string ; if ^ be intercepted, the tone will
be the tierce major.
But if this bridge only prevents the whole of the
string from vibrating-, without intercepting the commu-
nication of motion from the one part to the other, the
greater part will then emit only the same sound as the
less, and the | of the string, which in the former case
gave the fourth of the whole string, will give only the
double octave, which is the tone proper to the fourth
of the string. The case is the same if this fourth be
touched : its vibrations, by being communicated to the
other three-fourths, will make them sound, but in such
a manner as to give only the double octave.
The following reason, which may be rendered plain
by an experiment, is assigned for this phenomenon :
when the bridge absolutely intercepts all communica^
tion between .the two parts of the string, the whole of
the largest part vibrates together ; and if it l)e |, q( ^3e«^
u6
156 AMUSEMENTS
■whole string, it makes, agreeably to the general law, 4
vibrations in the time that the whole string would
make 3: its sound therefore is the fourth of the whole
string.
But, in the second case, the larger part of the string
divides itself into 3 aliquot parts, each of which is
equal to the le^s, and all -these distinct portions per-
form their particular vibrations; for if bits of red paper,
for example, be placed upon all the points of division,
and bits of white paper in the middle of each division,
the former will remain motionless, but the latter will
drop off as soon as the string begins to vibrate.
If the part of the string immediately made to vibrate,
instead of an aliquot part of the remainder, be only -^
of it, the whole string will then divide itself into se-
venth parts, and will emit only that tone which be-
longs to ^ of its length.
If the less part of the string be incommensuriable
to the greater, the sound will absolutely be discordant,
and will almost immediately cease.
METHOD OF ADDING, SUBTRACTING, MUL-
TIPLYING, AND DIVIDING CONCORDS.
It is necessary for those who wish to understand the
theory of music, to know what concords result from
two or more concords, either when added or sub-
tracted, or when multiplied by each other. For thi»
reason we shall give the following rules :
PROBLEM I.
To add one concord to another.
Express the two concords by the fractions which re-
present them, and then multiply these two fractions
together : that is to say, first the numerators and then
the denominators : the number thence produced, will
express the concord resulting from the sum of the-two
concords given.
ON MUSICAL STRINGS. 157
EXAMPLE I.
Let it be required to add the fourth and fifth together.
The expression for the fifth is |, and that for the
fourth I, the product of which is i\ =: i, being the
expression for the octave. It is indeed well known,
that the octave is composed of a fifth and a fourth.
EXAMPLE TI.
What is the concord arising from the addition of the third
major and the third minor f
The expression of third major is |, and that of the
third minor is |^, the product of which is |g or |, which
expresses the fifth ; and this concord indeed \% com-
posed of a third major and a third minor.
EXAMPLE III.
What is the concord produced bj/ the addition of tuo tones
' major i
A tone major is expressed by \ ; consequently, to
add two tones major, |- must be multiplied by |. The
product \\ is a fraction less than y^, or -f , which ex-
presses the third major ; hence it follows, that the con-
cord expressed by 4t> is greater than the third major,
and consequently two tones major are greater than a
third major, or form a third major false by excess.
On the other hand, by adding two tones minor, which
are each expressed by y^^, it will be found that their
sum ^'^ is greater than -j-^ or |, which denotes the
third major: two tones minor therefore, added toge-
ther, make more than a third major.
This third is, indeed, composed of a tone major aiid
a tone minor, as may be proved by adding together
the concords \ and ^, which makes JJ = -^ or |.
It might be proved, in like manner, that two semi-
toies miyor make nqore than a tonia.ma^yyc, ^sl^vsi^
] 53 AMUSEMENTS
semi 'tones minor less even than a tone minof ; and,
in the last place, that a semi-tone major and a semi-
tone minor make exactly a tone minor.
PROBLEH II.
To subtract one concord from another.
Instead of multiplying together the fractions which
express the given concords, invert that which expresses
the concord to be subtracted from the other, and then
multiply them together as before: the product will
^ve a fraction expressing the concord required.
EXAMPLE X.
What is the concord which results from the fifth subtracted
from the octave 1
The expression of the octave is J ; that of the fifth
i, which inverted gives |; and if J be multiplied by
I we shall have J, which expresses the fourth.
EXAMPLE II.
What is the difference between the tone major and the
tone minor ?
The tone major is expressed by |, and the tone
minor by ^, which when inverted gives *^ : the pro^
duct of |- by y* is 4?, which expresses the difference
between the tone major and the tone minor. This is
what is called the great comma.
PROBLEM III.
To double a concord, or to multiply it any number of times,
at pleasure.
'. In this case, nothing is necessary but to i«ii# the
wten&s of ti^e fittction, iirhich expresses the (iyea €xmh
ON MUSICAL STRINGS. 159
cord, to the power denoted by the number of times it
is to be multiplied ; that is, to the square if it is to be
doubled, to the cube if it be tripled, and so on.
Thus, the concord arising from the tone major tripled
18 -f^T > ^^r ^ ^^^ expression of the tone major is ^^
we shall have 8x8x8 = 512, and 9x9x9=;:
729. This concord |4f corresponds to the interval
between ut and a Ja, higher than fa sharp of ^he
g^amut.
PROBLEM IV.
To divide one concord by any number at pleasure, or to
find a concord which shall be the halj\ third, Sfc. of' a
given concord.
To answer this problem, take the fraction which
expresses the given concord, and extract that root of it
which is denoted by the determinate divisor; that is
to say, the square root, if the c'oncord is to be divided
into two; the cube root, if it is to be divided into
tWee, &c. ; and this root will express the concord re-
quired.
EXAMPLE.
As the octave is expressed by J, if the square root
of it be extracted, it will give ^ nearly ; but ^ is les»
than 4> &nd greater than y, consequently the middle
of the octave is between the fourth and the fifth, or
very near fa sharp.
OF THE RESONANCE OF SONOROUS BODIES,
THE FUNDAM!ENTAL PRINCIPLE OF HAR-
MONY AND MELODY, WITH SOxME OTHER
HARMONICAL PHENOMENA.
EXPERIMENT X.
If you listen to the sound of a bell, especially when
very grave, however indifferent your ear may be, yoa
wUl ecMiily distiiiguish; besides the priiifingaL f^y^a^^
160 ' AMUSfiMfiNTS
veral other sounds more acute; but if you have an eat
accustomed to appreciate the musical intervals, yoti
will perceive that one of these sounds is the twelfth
w fifth above the octave, and another the seventeenth
major or the third major above the double octave. If
your ear be exceedingly delicate, you will distinguish
also its octave, its double and even its triple octave :
the latter indeed are somewhat more difficult to be
heard, because the octaves are almost" confounded
with the fundamental sound, in consequence of that
natural sensation which makes us confound the octave
with unison.
If the bow of a violoncello be strongly rubbed against
one of its large strings, or the string of a trumpet ma-
rine, you will perceive the same effect. In a word, if
you have an experienced ear, you will be able to dis-
tinguish these different sounds, either in the resonance
of a string, or in that of any other sonorous body, and
even in the voice.
Another method of making this expeiiment.
Suspend a pair of tongs by a woollen or cotton cord>
or any other kind of small string, and twisting the ex-
tremities of it around the fore-finger of each hand, put
these two fingers into your ears. If the lower part of
the tongs be then struck, you will first hear a loud
and grave sound, like that of a large bell at a distance;
and this tone will be accompanied by several others,
more acute, among which, when they begin tp die
away, you will distinguish the twelfth and the seven-
teenth of the lowest tone. Rameau confirmed the
truth of this phenomenon by the help of several organ-
pipes.
This experiment respecting the resonance of sono-
rous bodies, is not new. It was known to Dr. Wallis
and to Mersesne, who speak of it in their works ; but
it appeared to them a simple phenomenon, with the
xonsequences of which they were entirely unacquainted.
Rameau first discovered the use of it, in deducing all
the. rule* ofjnusiQal composition, which befoteliad
ON MUSICAL TONES. 161
been founded on mere sentiment, and on experienccy
incapable of serving as a guide in all cases, and of ac-
counting for every effect. It forms the basis of his theory
of thorough bass ; a system which has been opposed with
much declamation, but which, however, most musicians
seem at present to have adopted.
All his harmony, therefore, is multiple, and composed
of sounds which would give the aliquot parts of the
sonorous body, }, y, J, |, ^, and we might add, -y, .
4, &c. But the weakness of these sounds, which go
on always decreasing in strength, renders it difficult to
distinguish them. Rameau, however, says, that he
could distinguish very plainly the sound expressed by
•J-, which is the double octave of a sound divided nearly
into two equal parts, being the interval between 7a and.
si flat, below the first octave ; he calls it a lost sound,
and totally excludes it from harmony.
£:!iCP£RIMlNT II.
If you adjust several strings to the octave, the
twelfth, and the seventeenth, of the determinate sound
emitted by another string, both ascending and de-
scending; as often as yc?u make that which gives the
determinate sound to resound strongly, you will im-
mediately see all the rest put themselves in a state of
vibration : you will even hear those sound which are
tuned lower, if you take care to damp suddenly, by
means of a soft body, the sound of the former.
Most people have heard the glasses on a table emit
a sound when a person near them has been singing
with a strong and a loud voice. The strings of an in^
strument, though not touched, are often heard to sound
in consequence of the same cause, especially after
swelling notes long continued.-
In a manner somewhat similar, the diversity of tones
agitates, in various ways, the fibres of our bodiei, ex*'
cites the passions, and produces in the soul sensations *
so different.
162 AMUSEMENTS
On the harmonical sounds heard with Ike principal sound :
xshether they have their source immediately in the lo*
narous body, or exist in the air or the organ f
It is very probable that the principal lound is the
only one that derives its origin immediately from the
vibrations of the sonorous body. Philosophers of emi-
nence have endeavoured to discover whether inde-
pendently of the total vibrations made by a body, there
are also partial vibrations ; but hitherto they have been
able to observe only simple vibrations. Besides, how
can it be conceived that the whole of a string should
be in vibration, and that during its motion it should
divide itself into two or three parts that perform also
their distinct vibrations ?
^ It must then be said, that these harmonical sounds
of octave, twelfth, seventeenth, &c. are in the air or
the organ : both suppositions are probable ; for such a
determinate sound has the property of putting into a
state of vibration bodies disposed to give its octave, its
twelfth, &c. we must allow that this sound may put
in motion the particles of the air, susceptible of vibra-
tions of double, triple, quadruple, and quintuple velo-
city. What, however, appears most probable in this
respect is, that these vibrations exist only in the ear :
it seems indeed to be proved by the anatomy of this
organ, that sound is transmitted to the soul only by
the vibrations of those nervous fibres which cover the
interior part of the ear ; and as they are of different
lengths, there are always some of them which perform
their vibrations isochronous to those of a given sound.
But, at the same time, and in consequence of the pn>-
perty above mentioned, this sound must put in motion
those fibres susceptible of isochronous vibrations, and
even those which can make vibrations of double, triple,
quadruple, &c. velocity. Such, in our opinion, is the
most probable explanation that can be given of this
singular phenomenon.
ON MUSICAL TONES. WSl
Of the modern music.
Every one knows that the gamut, or diatonic scale^
is. represented by the sounds ut, re, mi, fa, sol, la, si,uty'
which complete the whole extent of the octave ; and)
it appears, from the generation of it, as explained by
Rameau, that from ut to re there is a tone major;.'
from re to mi a tone minor; from mi to fa a semi-toiie;
major; from /a to sol a tone major, as well as iioia i<d)
to la; and in the last place, that from la to si there 'm
a tone minor, and from si to ut a semi- tone major;
It is thence, concluded, that in this scale there arier
three intervals which are not entirely just : these are,
1st. The third minor, from re to fa, which, beii».
composed of a tone minor and a semi-tone ms^or, k»
only in the ratio of 27 to 32; but this ratio isson^en
what less than that of 5 to 6, which expresses exa^il})
the third minor.
2nd. The third major, from/a t6 la, is too high, being .
composed of two tones major ; whereas, to be exactly
in the ratio of 4 to 5, it ought to consist of a tone
major and a tone minor.
3rd. The third minor, from la to tU, is as far from
being just as that of from re to fa, and for the same
reason.
On the came of the pleasure arising from music^^The
effects of harmony on man and on animals.
It has often been asked, why two sounds, which
form together, the fifth and the third, excite pleasure,
while the ear experiences a disagreeable sensation, by
hearing sounds which are no more than a tone or a
semi-tone distant from each other. Though it is diffi-
cult to answer this question, the following observations
may tend to throw some light upon it.
Pleasure, we are told, arises from the perception of
relations, as may be proved by various examples taken
from the arts. The pleasure, therefore, derived from
music, consists in the perception of the relations of
sound«. But are these relations sufficiently simple.for
164 AMUSfiMENtd
the loul to perceive and distinguish their order? Sounds
will please when heard together in a certain order;
bat, on the other hand, they will displease if their rela-
tions are too complex, or if they are absolutely desti-
tute of order.
This reasoning will be sufficiently proved by an enu- .
meration of the known concords and discords. In
nnisou, the vibrations of two sounds continually coin-
cide, throughout the whole time of their duration;
this is the simplest kind of relation. Unison also is
the first concord in the octave; the two sounds of
which it is composed perform their vibrations in such
a manner, that two of the one are completed in the
same time as one of the other : thus the unison is suc-
ceeded by the octave. It is so natural to man, that
he, who through some defect in his Voice, cannot reach
a sound too grave or too acute, falls into the higher
or lower octave.
- When the vibrations of two soundd ate performed in
such a manner, that three of the one correspond to
6ne of the other, these give the simplest relation next
to those above mentioned. Who does not know, that
the concord most agreeable to the ear is the twelfth,
or the octave of the fifth? In that respect it even sur-
passes the fifth.
• Next to the fifth, is the double octave of the fifth,
or the seventeenth major, which is expressed by the
ratio of 1 to 3. This concord, next to the twelfth, is
the most s^greeable.
The fourth, expressed by |, the third minor, ex-
pressed by -^j and the sixths, both major and minor,
expressed by ^ and |, are concords, for the same
reason.
But it appears that alhthe other sounds, after these
relations, are too complex for the soul to perceive their
order.
The, following very strong objection, however, may
be made to this reasoning. How can the pleasure
arising from concords consist in the perception of them,
since the soul often does not know whether such rela-
tions exist between the sounds? The most ignbrioid
ON MUSICAL TONES. 165
person is no less pleased with an harmonious concert
than he who has calculated the relation of all its parts;
what has hitherto been said, may therefore be more
ingenious than solid.
We cannot help acknowledging, that we are rather
inclined to think so; and it appears to us, that the
celebrated experiment on the resonance of bodies, may
serve to account, in a still more plausible manner, for the
pleasure arising from concords ; because, as every sound
degenerates into mere noise when not accompanied with
its twelfth and its seventeenth major, besides its octaves,
is it not evident, that when we combine any sound,
with its twelfth, or its seventeenth major, or with both
at the same time, we only imitate the process of nature,
by giving to that sound, in a fuller and more sensible
manner, the accompaniment which nature itself gives
it, and which cannot fail to please the ear, on account
of the habit it has acquired of hearing them together?
This is so agreeable to truth, that there ^re only two
primitive concords, the twelfth and the seventeenth
major ; aad that the rest, as the fifth, the third major,
the fourth, and the sixth are derived from them. We
know also, that these two primitive concords, are the
most perfect of all, and that they.form the most agree-
able accompaniment that can be given to any sound,
though on the harpsichord, for example, to facilitate
the execution, the third major and the fifth itself, which
with the octave, form what is called perfect harmony,
are substituted in their stead. But this harmony is
perfect only by representation, and the most perfect of
all, would be that in which the twelfth and the seven-
teenth were combined with tlie fundamental sound
and its octaves; Rameau, therefore, adopted it as often
as he could in his choruses. We might enlarge far-
ther oii this idea; but what has been already said will
be sufficient for every intelligent reader.
Some very extraordinary things are related in regard
to the effects produced by the music of the ancients,
which, on account of their singularity, we shall here
mention. We shall then examine them more minutely,
166 AMUSEMENTS
and shew that in this respect the modem music is not
inferior to the ancient.
Agamemnon, it is said, when he set out on the expe-
dition against Troy, being desirous to secure the fide-
lity of his wife, left with her a Dorian musician, who,
by the effect of his airs, rendered fruitless for a long
time, the attempts of iEgisthus to obtain her affection ;
but that prince having discovered the cause of her re-
sistance, got the musician put to death; after which
he triumphed, without difficulty, over the virtue of
•Clytemnestra.
We are told also, that at a later period, Pythagoras
composed songs or airs capable of curing the most
violent passions, and of recalling men to the paths of
virtue and moderation. While the physician prescribes
draughts for curing bodily diseases, an able musician
might therefore prescribe an air for rooting out a
vicious passion.
The story of Timotheus, the director of the music of
Alexander the Great, is well known. One day, while
the prince was at table, Timotheus performed an air
in* the Phrygian taste, which made such an impression
on him, that being already heated with wine, he flew to
his arms, and was going to attack his guests, had not
Timotheus immediately changed the stile of his per-
formance to the Sub-Phrygian. This change calmed
the impetuous fury of the monarch, who resumed his
place at table. This was the same Timotheus, who at
Sparta experienced the humiliation of seeing publicly
suppressed four strings which he had added to his lyre.
The severe Spartans thought that this innovation would
tend to effeminate their manners, by introducing a more
extensive and more variegated kind of music. This
at any rate proves, that the Greeks were convinced that
music had a peculiar influence on manners ; and that
it was the duty of government to keep a watchful eye
over that art.
Who indeed can doubt that music is capable of pro-
ducing siich an effect? Let us only interrogate our-
selves, and examine what have been our sensations on
ON MVSrCAL STRINGS. 167
hearing a majestic or warlike piece of music,^ or a ten-
der and pathetic air sung or played with expression.
Who does not feel that the latter tends as much ta
melt the soul and dispose it to pleasure, as the former
to rouse and exalt it ? Several facts in regard to the
modern music, place it in this respect on a level with
the ancient.
The modern music, indeed, has had also its Timo-
theus, who could excite or calm at his pleasure the most
impetuous emotions. Henry III. king of France, hav-
ing given a concert on occasion of the marriage of the
Duke de Joyeuse, Claudin le Jeune, a celebrated mu-
sician of that period, executed certain airs, which had
such an effect on a young nobleman, that he drew h\^
sword, and challenged every one near him to combat;
but Claudin, equally prudent as Timotheus, instandj
changed to an air apparently Sub-Phrygian, which ap-
peased the furious youth.
What shall we say of Stradella, the celebrated com-
poser, whose music made the daggers drop from the
hands of his assassins? Stradella having carried off
the mistress of a Venetian musician, and retired with
her to Rome, the Venetian hired three desperadoes to
assassinate him ; but fortunately for Stradella they had-
an ear sensible to harmony. These assassins, while
waiting for a favourable opportunity to execute their
purpose, entered the church of St. John de Latran,
during the performance of an oratorio, composed by
the person whom they intended to destroy, and were
so affected by the music that they abandoned their
design, and even waited on the musician to forewarn
him of his danger. Stradella, however, was not al-
ways so fortunate; other assissins, who apparently had
no ear for music, stabbed him some time after at Ge-
noa: this event took place about the year 1670.
Everybody almost has heard, that music is a cure
for the bite of the tarantula. This cure, which was
formerly considered as certain, has by some been con-
tested ; but, however this may be, Father Schott in
his works, gives the tarantula air, which appears to be
very dull, as well as that employed by the Sicilian
] 68 AMUSEMENTS
fishermen to entice the thunny fish into their nets, —
But it is probable, that fish are no great connoisseurs
in music.
Various anecdotes are related respecting persons
whose lives have been preserved by music effecting a
iort of revolution in their constitutions. A woman
being attacked for several months with the vapours,
and confined to her apartment, had resolved to starve
herself to death: she was, however, prevailed on, but
not without great difficulty, to see a representation of
the Sena Padrona, at the conclusion of which she
found herself almost cured ; and renouncing her me-
lancholy resolution, was entirely restored to health by
a few more representations of the like kind.
There is a celebrated air in Switzerland, called
Ranz des Vaches, which had such an extraordinary ef-
fect on the Swiss troops in the French service, that they
always fell into a deep melancholy when they heard
it ; Louis XIV. therefore, forbade it ever to be played
in France, under the pain of a severe penalty. We are
told of a Scotch air,, which has a similar effect on the
natives of Scotland.
Most animals, and even insects, are not insensible to
' the pleasure of music. There are few musicians, per-
haps, who have not seen spiders suspend themselves
by their threads in order to* be near the instruments.
We have several times had that satisfaction. We have
seen a dog, who at the adagio of a sonata, never failed
to shew signs of attention, and some peculiar sensa-
tion by howling.
The most singular fact, however, is that mentioned
by Burney, in his History of Music. This author re-
' lates, that an officer being shut up in the Bastille, had
permission to carry with him a lute, on which he was
an excellent performer; but he. had scarcely made use
of it for three or four days, when the mice issuing from
their holes, and the spiders suspending themselves from
the cieHng by their threads, assembled around him to
participate in his melody. His aversion to these ani-
mals made their visit at first disagreeable; and in-
duced him to lay aside this recreation; but be sooa
ON MUSICAL INSTRUMENTS. 169
was so accustomed to them, that they became a source
o( amusement.
We have learned from persons worthy of credit, now
in London, that during their residence in the Levant,
they have witnessed the influence of certain Greek
songs on the oxen, which the Greek farmers employ in
agriculture.
Those who have seen at Bartholomew fair, in Smith-
field, two elephants follow exactly the measure of the
tunes played at the entrance of the place where they
were kept, and humour all their variations, by the
motion of their head and trunk, will find no difficulty
in believing what BufFon has said respecting the singu-
lar taste of these animals for harmony.
In a word, without deciding whether the fables of
Amphion and Arion may not, in some measure, be
founded on truth, we know that the noisy sound of
trumpets, and the harmony of military instruments, ex-
cite the courage of soldiers, and the ardour of horses;
and the directors of caravans take care to be accom-
panied on their march,, by performers on different in-
struments, the music of which has such an effect on
their camels, that they are better enabled to sustain
the fatigue they must undergo in traversing the burning
desarts of Arabia or Africa.
OF THE PROPERTIES OF CERTAIN INSTRU-
MENTS, AND PARTICULARLY WINP INSTRU-
MENTS.
We are perfectly well acquainted with the manner in
which stringed instruments emit their sounds ; but er-
roneous ideas were long entertained in regard to wind
instruraentfe, such as the flute; for the sound was as-
cribed to the interior surface of the tube. The cele-
brated Euler first rectified this error, and it results
from his researches,
1st. That the sound produced by a flute, is nothing
else than that of the cylinder of air contained in it.
2nd. That the weight of the atmosphere, which com-
presses it, acts the part of the stretching weight.
I
170 AMUSEMENTS.
3rd. That the sound of this cylinder of air, is exactly
the same as that which would be produced by a string
of the same mass and length, extended by a weight
equal to that which compresses the base of the cy-
lUnder.
This fact is confirmed by experiment and oalcula*
tidn; for Euler found that a cylinder of air ot 7J Rhm-
laodish feet, at a time when the barometer is at a
mean height, must give c-soUut ; and such is nearly
the length of the open pipe of an organ which emits
that sound. The reason of its being generally made 8
feet, is because that length is required at those times
when the . weight of the atmosphere is greater.
Since the weight of the atmosphere produces, in re-
gard to the sounding cylinder of air, the sante effect
as- that produced by the weight which stretches a
string, the more the weight is increased^ the more will
the sound be elevated; it is therefore observed, that
during serene warm weather the 'tone of wind instru-
ments is raised; and that during cold and stormy
weather it is lowered. These instruments also become
higher in proportion as they are heated ; because the
mass of the cylinder of heated air becoming less,
"while the weight of the atinoi^phere remans unchanged,
the case is exactly the same as if a s^ng should be-
come less, and be still stretched by the same weight :
every body knows that such a strii>g would emit a
higher tone.
But as stringed instruments must become lower, be-
cause the elasticity of the strings insensibly decreases,
it thence follows, that wind and stringed instruments,
however- well tuned they may be tp each other, soon
become discordant.
A very singular , phenomenon is observed in regard
t6 wind instruments, such as the flute and huntsman's
horn. With a flute, for example, whep all the holes
are stopped, if you blow faintly into the mouth aper-
ture, a certain tone will be produced; if you blow a
little stronger, the tone instantly rises to die octave,
and by blowing successively wi1:h more forc€> you will
produce the twelfth or fifth above the octave; theii
the double octave or seventeenth major. -
ON MUSICAL INSTRUMENTS. 171
Of same musical instruments or machines^ remarkable for
their singularity or construction*
At the head of all these music&l instruments, 0t
machines, we ought doubtless to place the organ ; the
extent and variety of the tones of which would eiu^il^
nuch more admiration, if it were not so common as it
is in our churches ; for, besides the artifice necesssyi^
t6 produce the^ tones by means of keys, what ingenui'Cy
must have been required to contrive mechanism Wt
giving that variety of character to the tones obtaincid
by means of the different stops, &c. ? A complete de-
scription, therefore, of an organ, and of its constiUctidA,
would be sufficient to occupy a large volume.
The ancients had hydraulic organs, that \^ to ^eiy,
organs the' soiund of which was occasioned by air prd«
duced by the motion of wat^. These maschines w«t^
inveitted by Ctesibus of Alexandria, and his schohftr
Hero. From the description of these hydraulic ot^^AK!,
given by Vitruvius, in the tentli book of his serchit^^ie^
ture, Perrault constructed dm^ v^hich he deposited ilL
the king's library, where the Royal Academy of Scien«fefc
held their sittings. This instrument, indeed, is not to
be compared to the modem organs ; but it is evidcSttt
that the mechanism of it has served as a basis for that
of ours. St. Jerome speaks with enthusiasm of an
organ which had twelve pair of bellows, and which
€Ould be heard at the distance of a mile. It thence
appears, that the method employed by CtesibtHH tJ6
produce air to fill the wind*box, was soon laid aside
for one mor^ simple ; that is to say, for a pair of bd*
lows.
The performer on the tambour de basqtie^ and th«^^ ati^-
tomaton flute-player of Vaucanson, which weire exhi-
bited and seen with admiration in most parts of £arQ{M,
in the year 1749, may be classed among the most cu-
rious musical machines ever invented. We shall ifoi;
however, say any thing of the former of these machines,
because the latter appears to have been far more cottfr
plex.
|2
172 AMUSEMENTS
The automaton flute-player performed several airs
on the flute, with the precision and correctness of
. the most expert musician. It held the flute in the
usual manner, and produced the tone by means of its
mouth; while its fingers, applied on the holes, pro-
duced the different notes. It is well known, how the
fingers might be raised by spikes fixed in a cylinder,
80 as to produce these sounds ; but it is difficult to be
conceived how that part could be executed which is
performed by the tongue, andwithout which the music
would be very defective. Vaucanson, indeed, confes-
ses, that this motion in his machine was that which cost
him the greatest trouble.
A very convenient instrument for composers, invented
in Germany, consists of a harpsichord, which, by cer-
tain machinery added to it, notes down any air while a
person is playing it. This is a great advantage to com-
posers, as it enables them, when hurried away by the
fervour of their imagination, to preserve what has suc-
cessively received from their fingers a fleeting exist-
ence, and what otherwise it would often be impossible
for them to remember. A description of this machine
may be found in the Memoirs of the Academy of Berlin
for the year 1773.
Of a new instrument called the Harmonica.
This instrument was invented in America by Dr.
Franklin, who gave a description of it to father Bec-
caria, which the latter published in his works, printed
in 1773.
It is well known, that when the finger, a little mois-
tened, is rubbed against the edge of a drinking glass,
a sweet sound is produced-; and that the tone varies
according to the form, size, and thickness of the glass.
The tone may be raised or lowered also by putting
.into the glass a greater or less quantity of water. Dr.
Franklin says, that an Irishman, named Puckeridge,
^st conceived the idea, about twenty years before, of
constructing an instrument with several glasses of this
kind, adjusted to the various tones, and fixed to a
ON MUSICAL INSTRUMENTS. 173
Stand in such a manner, that different airs could be
played upon them. Mr. Puckeridge having afterwards
been burnt in his house, along with this instrument^
Mr. Delaval constructed another of the same kind, with
glasses better chosen, which he applied to the like
purpose. About fourteen or fifteen years ago, an
English lady at Paris, performed, it is said, exceed-
ingly well on this instrument, which, however, did not
long continue in vogue : at present it is confined to
cabinets and other musical curiosities.
A juggler, some years ago, to shew his dexterity,
placed on a table eight glasses of the same size, which
had all the same tone, and boasted that he could tune
them in an instant by pouring water into them, so as
to play an air with the utmost precision. ''Those
who tune violins or organs, (said he) cure not so dex-
terous as I; since they often labour for a quarter of an
hour, and try the same pipe or string twenty times,
before they can bring it to the proper tone.** While he
pronounced these words he poured water into the
eight glasses ; then striking them one after the other
with a small rod, he immediately shewed that they
emitted with great exactness the tones of the gamut,
ut^ re, miy fa, sol, la, si, ut ; and as he then amused
the spectators by playing an air, which he accompa-
nied with his voice, they overlooked the artifice he had
employed in tuning his instrument so speedily.
Each of the glasses had a small hole at the proper
height, so that when filled to the brim the water ran
out, tin there remained no more than the quantity re-
quisite to give the glass the necessary. tone. By these
means, the instrument tuned itself in a moment ; and
the musician had no occasion to pour in or pour out
water, at different times, to render the tone graver or
more acute.
On what ts~ called a false voice »
A fine voice is certainly preferable to every instru-
ment whatever. Unfortunately, many persons have
only a fafce voice; but, in general, this does not arise
I 3
174 AMUSBMfiNTg
from any defect in the organs of the voice, which are
aimoflt the same in ail mankind. It originates from the
ears, owing to an inequality of strength in these organs,
or to some want of delicacy or tension, in consequence
oi which, as they receive unequal impressions, we ne^
oessarily hear false sounds, and the voice, which en-*
deavours to imitate them, becomes itself false. On
this subject Df . Vandennonde made a very simple ex-
periment, which he relates in his Essay on Improving
the Human Mind, and which may be repeated on chil-
dren who pronounce with a false voice, in order that a
remedy may be applied in that tender age, when the
Ofg^ns are still susceptible of modification.
The experiment as he describes it, is as follows :—
*^ I made choice (says he) of a clear day, and having
fised on a spacious apartment, I took up my station
in a place judged most convenient for my experiments.
I then stopped one of the ears of the chttd who was to
ha the subject of them, and made her recede from me,
till she no longer heard the sound of a repeating watch
which I held in my hand, or at least until the sound
of the bell produced a very weak impression on her
organs of hearing. I then desired her to remain in
that place, and immediately going up to her unstopped
her ear, and stopped the other, taking care- to cause her
to shut her mouth, lest the sound should be commu-
nicated to the ear through the eustachian tube. I
then returned to my station, and making my watch
again strike, the child was quite surprised to find that
she heard tolerably well ; upon which I made a sign to
her to recede again till she could scarcely hear the
sound.**
It results from this experiment, that in the ears of
persons who have a false voice, there is an inequality
of strength, and the means of remedying this defect in
children, is to ascertain by a similar mode, which ear
is the weakest.
** When this has been discovered, nothing better can
be done, in my opinioix, (says Dr. Vandermonde) than
to stop up the other as much as possible, and to take
advantage of that valuable opportunity of frequently
ON MUSICAL INSTRUMENTS. 175
exercising the weak ear, but in such a manner as not
to fatigue it. The one thus made to labour alohe'will
always retain the same force. The child's ear should
from time to time be unstopped, in order to make it
sing, and to discover whether both ears have the same
degree- of sensibility."
This natural defect may be then corrected, and any
person may be made to acquire a true voice, provided
the means pointed out by Dr. Vandermonde be early
employed.
Persons who have a false voice, in consequence of
8ome inequality in the ears, may be compared to those
who squint: that is to say, who, in order to see an ob-
ject distinctly, do not turn equally towards it the atili
of both 6ye8, because they have \ not the same visual
powers. It is probable, that the former, if they had
early accustomed themselves to make use of only one
ear, would hear distinctly different sounds which the^
would have imitated^ and would not have contracted a
false voice.
Of the Speaking Trumpit and Ear Trumpet.
As the sight is assisted by telescopes and microscopes,
similar instruments have been devised also for assisting
the faculty of hearing. One of these, called the speak-
ing trumpet, is employed for conveying sound to a
great distance : the other, called the ear trumpet, serves
to magnify to the ear the least whisper.
Sir Thomas Mori and, among the modems, bestowed
the most labour in endeavouring to improve this me-
tfiod of enlarging and conveying sound ; and on this .
subject he published a treatise, entitled De Tubd ten^ ^
torophonica, a name which alludes to the voice of sten-
tor, so celebrated among the Greeks for its great
strength. The following observations on this subject
are in part borrowed from that .curious work.
The ancients, it would seem, were acquainted with
the speaking trumpet, for we are told that Alexander
had a horn, by means of which he could give orders to
his whole army, howetver numerous. l^o^^T^^xt^*^^
i4
176 AMU&EMCMl*!? .
authority of some passages in a manuscript, preserved
in the Vatican, makes the diameter of its greatest
aperture to have been seven feet and a half. Of its
length he says nothing, and' only adds, that it could
be heard at the distance of 500 stadia, or about 25
miles.
However this may be, the speaking trumpet is
nothing else b\it a long tube, which at one end is only
large enough to receive the mouth, and which goes on
increasing in width to the other extremity, bending
somewhat outwards. The aperture at the small end
must be a little flattened to fit the mouth; and it
ought to have two lateral projections to cover part of
the cheeks.
Sir Thomas Morland says, that he caused several
instruments of this kind to be constructed, of differ-
ent sizes, viz. one of four feet and a half in length, by
which the voice could be heard at the distance of 500
geometrical paces; another 16 feet 8 inches, which
conveyed sound 1800 paces; and a third, of 24 feet,
which rendered the voice audible at the distance of
2500 paces. •
The reason of this phenomenon is as follows : — As
the air is an elastic fluid, so that every sound pro-
nounced in it is transmitted spherically around the so-
norous body, when a person speaks at the month of the
trumpet, all the motion which would be communicated
to a spherical mass of air, of four feet radius^, for
example, is communicated only to a cone, the base of
which is the wider extremity of the trumpet. Conse-
quently, if this cone is only the hundredth part of the
whole sphere of the same radius, the effect will be
the same as if the person should speak a hundred times
as loud in the open air : the voice must therefore be
heard at a distance a hundred times as great.
The ear trumpet, an instrument exceedingly useful
to the deaf, is nearly the reverse of the speaking trum-
pet : it collects, in the auditory passage, all the «ound
contained within it; or it increases the sound produced
at its extremity, in a ratio which may be said to be. as
that of the wide end- to the narrow end. Thus, for
ON MUSICAL INSTRUMENTS. 177
example, if the wide end be 6 inches in diameter, and
the aperture applied to the ear 6 lines, which in sur-
faces gives the ratio of 1 to 144, the sound will be in-
creased 144 times, or nearly so; for we do not believe
that this increase is exactly in the inverse ratio of the
surfaces ; and it must be allowed, that in this respect
acoustics are not so far advanced as optics.
t 5
178 jUtUSEMERTS
ELECTRICITY.
DEflNITIOKS.
1. ELECTRICITY is that property in bodies which
enables them, when excited by friction or beat, to at-
tract other light bodies, and produce an effluvium that
is sometimes luminous, attended with a snapping noise,
and a faint phosphoreal smell.
2. Electricity is called the' second of the three species
of attraction, gravity being the first, and magnetism the
third.
3. Those bodies that produce electricity by friction
or heat, are called electrics, and are &aid to be electric
per se,
4. Those bodies that receive and communicate elec-
tricity are called conductors, and those that repel or
will not suffer it to pass through them, are called non-
conductors.
5. All bodies that are made to contain more than
their natural quantity of electricity are said to be elec-
trified positively, and those from whom part of theit
natural quantity is taken away, are said to be electri-
fied negatively. These two electricities being first pro-
duced, one of them from glass, and the other from am-
ber, wax, or rosin, the former was called vitreous and
the latter resinous electricity.
6. When a quantity of electricity is communicated
to any body, it is said to be charged.
7. The effect of the explosion of a charged body,
that is, the discharge of its electricity through any other
body, is called the electric shock.
' 8. When any body is prevented from communicating
with the earth, by the interposition of an electric body,
it is said to be insulated*
9. The residuum oto. charged body> as ajar or bat-
tery, is that part of the charge which remains in the
body after the firiit discharge, and by which it will give
a second shock, though leds than the first.
■
APHORISMS.
1. All substances are distinguished into electrics per
sty and non electrics: the latter of which are conduc-
tors, and the former non-conductors.
2. All kinds of metals, seml-metals, water, charcoal,
and other bodies of a similar nature, are conductors ;
and all other bodies, whether minei'al, vegetable, o*
animal, are non-conductors: many of the latter, how-
ever, may be made to conduct electricity by being
heated to a certain degree.
3. Positive electricity is produced by the friction of
uninsulated glass tubes or globes ; ana negative elec-
tricity is produced either from the rubber of those bo-
dies, or from the friction of insulated glass bodies ; or
lastly, frdm the rubbing of globes or sticks of wax, sul-
phur, and other bodies of a similar nature;
4. It follows from the last aphorism, that the .elec-
tricity of the excited body and the rubber, are always
opposite, that is, if that of the excited body be posi-
tive, that of the rubber will be negative: and the con-
trary. Those two bodies moreover \viU act (m each
other with greater force than axty other body.
6, In charging any body, a^ a coated pm£l}, if one
side communicate with the excited body, and the other
with the rubber, the el^tricity of the two sides of the
charged body will be opposite.
6. There is- a strong attraction between the t#p elec-
tricitieB oh tii^ opposite stde^ of a gl^ss, s6 that when
they are' lAade to comtouAitiate by means of A cdndoc-
tor, di^ #f(» he bdth disdhargtid wrth ft flUh <«Pli^:
and a stiftppitlg noi^e. '
7« The substance of glass is iim>e!fVidMi t6 electl'i-
city ; but if l*i* glAiA be thte, and the elettricity on the
cfpii$m iUtel U i^smiAg, ftat is, if the ^aA«L V^^
I 6
180 AMUSEMENTS
overcharged, the opposite electricities will force a paS"
4age through the glass.
8. If an excited electric be in contact with an insu*
lated conductor, the former will communicate its power
to the latter, which will then attract light bodies, and
give a spark) in the same manner as the excited elec-
tric.
9. The flash of light from a body to which electri-
city has been communicated, is more dense, and the
sound louder, than from one that is excited ; for the
conductor parts with all its electricity at once, but the
excited body with only so much as is at or near the
part that is touched.
10. If insulated bodies have been attracted by, and
have. touched an excited body, they will soon after
be repelled by that body, and will repel each other ;
nor will they return to the excited electric till after they
have touched some other body that communicates with
the earth.
1 1. When an insulated conductor is brought within
the sphere of action with an excited body, it acquires
the electricity opposite to that of the body, and the
nearer it is brought, the greater quantity it acquires,
till.^tl^e Que receive a spark from the other, and then
tlie^^lc;QtrlcitY of both is discharged.
1,2. The electric explosion always takes the shortest
course. through the best conductors.
\ 13. If the explosion between two bodies be inter-
rupted by a non-conductor of a moderate density, the
. discharge will force a passage through it^ in such a
mapner as to leave the appearance of a sudden expan-
sion of the air about the center of the ei^plosioil.
14. If an insulated conductor be pointed^; or if an
upji^sulated conductor that is pointed, be brought very
near the earth, there will be no other appearance of
electricity during the time of excitation than a light,
aVid a current of air, that may be perceired to cooae;
from.t(iose points, . > ;* ..
15. The electric attraction acts in vacuo[. ii \,y\ ; / ^
16. Electricity an4 lightoing are in i^i r^q^^^jiir.a*
similar nature.
IN ELECTRICITV. 181
17. All the effects of lightning may be imitated -by
electricity, and all the experiments in electricity may
be performed by lightning, brought down from the
clouds by means of an insulated pointed rod of metal^
or by a kitCr
Among the wonderful discoveries of human nature,
there is hardly any that rank higher than electricity.
This phenomenon, like many others, was found
out merely by accident: yet it has proved not only a
source for various experiments, but likewise extremely
beneficial to mankind.
The great Dr. Franklin has improved more in this
branch of knowledge than any other person. He even
contrived to bring lightning from the clouds by means
of conductors : — these conductors are of great service,
when fixed to churches, and other public edifices, to
preserve them from the dreadful effects of the rapid-
ness of elemental fire.
When electricity is made use of physically, it is of
great utility, and has been known to relieve, and some-
times entirely cure, various disorders. It is very ser-
viceable in the rheumatism, and other chronic dis-
orders.
** One circumstance," says Mr. Gale, in his Recrea-
tions, " I shall mention, which I received from a gen-
tleman who has been dead some years, but whose cha-
racter as an artist and an ingenious person, will be a
long time remembered ; I mean Mr. Benjamin Rack-
strow, of Fleet Street.
** He' told me, that having some company one day to
see his museum, and his electrical experiments, they
were rather fearful of undergoing the shock ; when a
person who was much given to inebriety, being in the
room, and rather intoxicated, voluntarily offered to let
the experiment be tried on him : this was agreed to,
upon which he received it pretty smartly three or four
times, and thought no more about it at that time. A ^
fbv '^B.y» Afterwards he had occasion to go to Chiches*
t^, tik SuiBfiex/'and being rather low ipi circumstances^
wiikt)bfi*dd;teWflk.
] 8S AMUSEMENTS
** This man had been affected for many yeftrs with a
rupture, which was extremely troublesome ; but on his
journey he had not the least symptom of it : on which
he wrote a letter to Mr. Rackstrow, informing him of
this agreeable circumstance, and imputing it entirely
to his receiving the shock from his electrical appara-
tus. The man lived to confirm this by word of mouth j
and what is really extraordinary, the rupture never
returned : which is sufficient to establish its physical
consequence. It is of farther service in palsies and
contractions, and is performed by sparks, drawn by
firiction, from the electrical machine.
" Its real use being thus established, we may now,
without offending, be a little merry with other circum-
stances which have and may happen again, by means
of electricity.
** Some ladies and gentlemen, coming to Mr. Rack-
5trow's, brought with them a negto servant who had
pot been long in England : after they had seen his na-
tural and artificial curiosities, they desired to see some
of his electrical experiments, and gave him a hint to
play a trick or two upon poor Mungo. Mungo was
not a little surprised at the shocks he received, but
could not guess from whence they came ; but when the
rooirn was darkened, and fire made to come out of
his fingers' ends, he roared out like a mad bull, crying
the devil ! the devil ! and in endeavouring to get out
of the room, overset the skeleton of a rhinoceros^ run
his head against a case of butterflies, and broke to
pieces a fine bust of the Marquis of Granby ; and hav-
mg once more gained day-light, made a sudden spring
into the street, and run immediately home, to the no
•mall: diven^oti of his master and family.
** Mrs. Bulky being troubled with a tympany, wa«
febctmmended to be eie^tHfied ; she accordingly went
to a professor* in that way, who asked her if .dbe could
beaf a pretty hard shock. O yes, sir, said ske, as hard
us you please, ^td tiS often as you please ; I am jrtf]^
fcfdi of being shocked, the man by this supposoa die*
YsbA tfdbre ntid^tgdiiier t^^ bfieftition^ ^^'^ zT!' ^Vf WF>
tng to give her what »he seemed %<> well t8 ttn%r*
IN ELECTRICITY. 183
Stand ; but, alas ! he wound up his instrument too
high ; so that he not only overset his patient, bat ac-
tually conveyed her into a cellar where they sold ox
cheek and peas-soup; down went the streaming
pan full of savory broth, and off flew her monument
of a cap into the other boiling cauldron.
'* The cook reddened like a heated poker, the cus-
tomers rose from their seats, and the greatest confusion
took place in this subterraneous abode.
'' All culinary business was at an end for the present^
the electrical doctor came running to the assistance of
his patient; but as soon as the cause of the disaster
was explained, the occupier of the place declared the
damages should be made good, her pan of leg of bee^
was entirely lost, her peas soup spoiled by the powder
and pomatum of the lady's head dress, the doctor wad
the cause of all, and he should pay. for all ; but he de-
clared, he would, sooner than pay a farthing, eleciiiff
the house till it fell about their ears.
*' At last the lady, having adjusted herself in the best
manner she could, gave the good woman a crown, and
so compromised the matter ; however, it cured her of
her tympany, for she never went to the doctoy after<»
wards.
** Many are the tricks played>bympans of an electri-
fying machine. A person in London had one in his
i^op, which was not seen by the passers by; and he
hung at the door an old steelyard, which^ from its
make, seemed to be very ancient ; this attracted the
attention and notice of many, who no sooner went to
examine it, than they received the shock ; those that
knew what it was, only smiled and went on ; others
stared,^ and could not guess from whence it cam^
** A <&uaken porter being called one day, and asked
whsEt he would have to carry the steelyard ta a ceEtain
place, went to examine it ; but he no ivooner* touehed it
tfarSKhe-felt a blow, and turning' roimdv with anoeitb
declared^, if h^« knew wbcr it was, he would pity ^em
weH for their impudence; He'tkeo: returned te- mpetik
fl^KHit bav jeby. ami recevred axiodier sboek, andaiiotker
aftarillMl?> titt^ittilated fay the<mippo»edi ataiPditSy^B^^f^
184 AMUSfiMCKTS
by he could not tell whom, he stripped to the buff in
order to fight all that came in his way, till he got a
mob of boys and dogs at his heels, and was glad to
get away at any rate. - >
. . ** Such tricks are not recommended as proper to be
practised, for they are really dangerous. A strange
person might, on findinjg the truth, break the windows,
or;keep it in his mind, and do the electryfying gentle*
man some injury, which might make him repent of his
experiments.
" Small electrical machines are often introduced in
company, and create not only mirth, but produce real
rational amusement ; such can never ba disagreeable,
l>ut must give satisfaction to all who have any idea of
philosophical knowledge, and wish to improve their
■minds by mathematical experiments ; to all such, we
may safely recommend the electrical apparatus, which
will be both useful and profitable.**
A description of all the machinery that has been used
in electrical experiments would fill a volume. We
therefore refer the reader to the numerous and labo*
rious productions on the subject of electricity, where
he may meet with ample descriptions of such appara-
tus, and hasten to detail some of the amusing experi-
ments in this science.
We have divided the following amusements into such
as are performed in the light, and such as require a dark
chamber; beginning with the former.
Tht Animated Feather*
Electrify a smooth glass tube with a rubber, and hold
a small feather (or piece of leaf gold) at a short dis-
tance from it. ' The feather will immediately fly to the
tube, and adhere to it for a short time, and then fly
off, and the tube can never be brought close to the
feather till it has touched the side of the room, or some
other body. that communicates • with the ground. If,
therefore, the operator take care to keep the ti;be con-
stantly between the feather and the sida of the room,
he may dnve it round to all parts without touching it;
IN ELECTRICITY. 185
&nd what is very remarkable, the same side of the
feather will be constantly opposite the tube.
While the feather is flying before the smooth tube,
it will be immediately attracted by an excited Jfough
tube, or a stick of wax, and fly continually from one
tube to the other, till the electricity of both is dis-
charged.
Tliis was one of the first, and is one of the most com-
mon experiments in electricity ; it is, however, very
entertaining, and shows the nature of electric attraction
and repulsion altogether as well as a more elaborate
performance.
The Maroellovs Fount ain*
Suspend a vessel of water from the middle of the
trass arch, and place in the vessel a capillary syphon*
The water will at first issue by drops only, from thet
lower leg of the syphon, but when the vessel is put in
motion, there will . be one continued stream of water,
and if the electrification be strong, a number of streams
will issue, in form of a cone, the top of which will be
at the extremity of the tube. This experiment may be
stopped and renewed, almost instantly, as if at the word
of command.
The Magic Picture.
Have a large print, suppose of the king, with a frame
and glass. Cut a panel out of the print at about
two inches from the frame all round ; with thin paste
or^um, fix the border that is cut off*, on the inside of
the glass, pressing it smooth and close, then fill up
the vacancy, by covering the glass well with leaf gold,
or thin tin-foil, so that it may lie close. Cover, like-
wise, the inner edge of the bottom part of the back of
the frame with the same tin-foil, and make a commu-
nication between that and the tin-foil in the middle of
the glass; then put in the board, and that side is fi-
nished. Turn up the glass and cover the foreside with
tin-foil, exactly over that on the backside, and when
it is dry, paste over it the pannel of the print that was
cat out, observing to bring the corresponding parts of
186 AMtrlSEMENtSt
the border and pannel together, so that the picture
will appear as at first, only part of it behind the glass,
and part before. Lastly, hold the print horizontally
by the top, and place a little moveaole gilt crown on
the king's head.
Now if the tin-foil on both sides of the glass be mo-
derately electrified, and another person take hold of
the bottom of the frame with one hand, so that his
fingers touch the tin-foil, and with the other hand en-
deavour to take off the crown, he will receive a very
smart blow, and fail in the attempt. The operator,
who holds the frame by the upper end, where there is
no tin-foil, feels nothing of the shock, and can touch
the face of the king, without danger, which he pretends
to be a test of his loyalty. When a ring of persons
take a shock among diem, the experiment is called the
conspirators.
The Tantalian C^i^.
Place a cup or pot, of any sort of metal, on a stool
of baked wood, or a cake of wax. Fill to the brim
with any sort of liquor; let it communicate with the
branch by a small chain, and when it is moderately
electrified, desire a person to taste the liquor, without
touching the cup with his hands, and he will imme-
diately receive a shock at his lips; which, however,
should not be very strong.
The motion of the wheel being stopped, you offer to
taste the liquor yourself, and desire the rest of the
company to taste it likewise, which they will do with-
out any inconvenience. You then gi?e the signal to
the operator, and while you are amusing the company
with discourse, the cup is again charged, and you de-
tire the same person a second time to taste the liquor^
when, to the no small diversion of the company, he will
receive a second shock.'
The Self-moting Wheel.
This wheel is formed of a thin round plate of win-
dow-glass, 17 inches diameter, well gilt on both sides^
all but two inches next the ^%i^. Two smaH hemi-
IN ELECTIUCITY. 187
spheres of wood are then fixed with cement tci the
middle of the upper and under sides, centrally oppo-
site, and in each of them a thick strong wire, eight of
ten inches long, which together make the axis of th^
wheel. It turns horizontally, on a point at the lower
end of its axis, which rests on a bit of brass cemented
within a glass salt-cellar. The upper end of its axis
passes through a hole in a thin brass plate, which keeps
It six or eight inches distant from any non-electric, a«d
has a small ball of wax or metal on the top, to keep
in the fire.
In a circle on the table which supports the wheel,
are fixed twelve small pillars of glass, at about eleven
inches distance, with a thimble on the top of each. On
the edge of the wheel is a small leaden bullet, commii*
uicating by a wire with the gildings of the upper sur-
face of the wheel ; and about six inches from it is ano-
ther bullet, communicating, in like manner, with the
under surface. When the wheel is to be charged by
the upper surface, a communication must be made from
the under surface to the table.
When it is well charged it begins to move. .The
bullet nearest to a pillar moves towards the thimble on
that pillar, and passing by, electrifies it, and then pushes
itself from it. The succeeding bullet, which commu-
nicates with the other surface of the glass, more strong-
ly attracts that thimble, on account of it being electri-
fied by the other bullet, and thus the wheel increases
its motion,^ till it is regulated by the resistance of the
air. It win go half an hour, and make, one minute
with another, 20 turns in a minute, which is 600 turns
in the whole. The bullet of the upper surface gives
in each turn 12 sparks to the thimbles, which makes
7200 sparks ; and the bullet of the under surface re*
ceives as many from the thimbles, those bullets moving
in the same time 2500 feet. The thimbles are well
fixed, and in so exact a circle, that the bullets may
pass within a very small distance of them.
If, instead of two bullets, you put eight, four com-
municating with the upper surface, and fouf with the
under surface, placed alternately (which eight, at about
188 AMUSEMENTS
six inches 'distance, Complete the circumference) the
force and celerity will be greatly increased ; the 'wheel
making 50 turns in a minute : but then it will not con«
tinue 60 long in motion.
Tke MagtciatCs Chace*
On the top of a finely painted wire, rising perpendi^*
cularly fram^the conductor, let another wire,- sharpened
at each end, be made to move freely, as on a center.
If it be well balanced, smd the points be bent horizon*
tally, in opposite directions, it will, when electrified,
turn very swiftly round, by the re-actiqn of the air
against the current which flows from off the points.
These points may be nearly concealed, and the figures
of men and horses, with hounds and a hare Or fox,
may be placed upon the wires so as to turn round -with
them, when they will look as if the one pursued the
other. If the number of wires proceeding from the
same center be increased, and a still greater variety of
■wires proceeding from the same center be increased^
and a still greater variety of figures be put upon them,
the chace must be more diversified and entertaining.
If the.wire which supports the figures have another
wire finely pointed, rising from its center, a second set
of mires, furnished with another sort of figures, may
be made to revolve above the former, and either in the
same or the contrary direction, as the operator shall
think, fit.
If such a wire, pointed at each end, and the ends
bent in opposite directions, be furnished like a dipping
needle, with a small axis fixed in its middle, at right
angles with the bending of the points, and the same be
placed between two insulated wire strings, near and
parallel to each other, so that it may turn on its axis,
fireely upon and between them, it will, when electrified,
have a progressive as well as circular motion, from
one end of the wires that support it to the cTther; and
this even up a considerable ascent.*
IN ELECTRICIlY. 189
The Planetarium »
From the branch suspend six concentric hoops of
metal, at different distances from each other; and
under them, on a stand, place a metal plate, at the
distance of about half an inch. Then place upon the
plate, ivithin each hoop, and near to it, a round glass
bubble, blown very light. These bubbles, and the dis-
tances between the hoops should correspond to the dif-
ferent diameters of the planets, and those of thdr or-
bits ; but as that cannot be, on account of the vast
disproportion between them, it must suffice here to
make a difference that bears some relation to them.
Now, the hoops being electrified, the bubbles, placed
upon the plate, near the hoops, will be immediately
attracted by them ; in consequence of which, that part
of a bubble which touches a' hoop will acquire some
electric virtue, and be repelled. The electricity not
being diffused over the whole surface of the glass, an-
other part of the surface will be attracted, while the
former goes to discharge its electricity upon the plate.
This will produce a revolution of the bubble quite round
the hoop, as long as the electrification is continued^ and
will be either way, just as the bubble happens to set
out, or is driven by the operator. A ball hung over
the center of all the hoops will serve to represent the
sun in the center of its system. If the room be dark-
ened the several glass balls will appear beautifully il-
luminated. This experiment affords a remarkable in-
stance of electric attraction and repulsion.
The Incendiaries,
l^ei a person stand upon a stool made of baked wood,
or upon a cake of wax, and hold a chain oommuni-*
eating with the branch. Upon turning the wJieel, he
will soon be electrified; his whole body, in reality,
making a part of the prime conductor, and will exhibit
the same appearances ; emitting sparks wherever he is
touched by any person standing on the floor. If the
prime conductor be very large, the sparks may be rather
190 AMUSEMENTS
painful than agreeable; bat if it be small , the electrifi-
cation moderate, and none of the company touch the
eyeSy or the more tender parts of the face, the experi-
ment is diverting enough to a,n parties.
Many of the preceding experiments may be also per-
formed to advantage by a person standing upon the
4itool as above, and holding in his hand what was di-
rected to be fastened to the prime conductor. If he
hold a large plumy feather in his hand, it is very pleas-
ing to observe how it becomes turgid, its fibres eirtend-
ing themselves in all directions from the rib ; and how
it shrinks like the sensitive plants, when any unelec-
trified body touches it; when the point of a needle is
presented to it, or to the prime conductor with which
he is connected.
If a dish, containing spirits of wine made warm, be
brought to the electrified person, and he be directed to
put his finger, or a rod of iron into it, the spirit will be
immediately in a blaze ; and if there be a wick oir thread
in the spirit, that communicates with a train of gun-
powder, he may oe made to blow up a magazine, or
set a city on fire with a piece of cold iron; and at the
same time know nothing of what he is about.
An amusement of this sort may be performed by se-
veral persons, standing upon insulated stools, and many
diverting circumstances may be added to those here
mentioned. Care should be taken that the floor on
which the stools stand be free from dust, but it is most
eligible to have a large smooth board for that pur-
pose.
The Inconceivable Shock,
Put into a person s hand a wire that is fixed on to
the hook that comes from the chain which communi-
cates with one side of the battery, and in his other hand
put a wire with a hook at the end of it, which you
direct him to fix on to the hook that comes from the
Other chain, which when he attempts, he will instantly
receive a shock through his body, without being able
to guess from whence it proceeds. The shock will b%
IN EI^ECTEICITY. 191
in proportion to the number of jars that are charged ;
but it is remarkable, that ar small shock gives a much
more pungent sensation in passing through the body^
than one that is large.
This amusement may be diversifiedy and rendered
still more entertaining, by concealing the chain that
communicates with that which comes from the outside
of the battery, under a carpet, and placing the wire
that communicates with the chain which comes from
the inside, in such a manner that a person shall put hip
hand upon it without suspicion, at the same time that
his feet are upon the other wire. Many other methods
of giving a shock by surprise may be easily contrived ;
but great care should be tdcen, that these shocks be
not too strong, and that they be not given to all per-
sons indiscriminately. ,
When a single person receives a shock, the company
is diverted at his sole expence ; but all contribute their
share to the entertainment, and all partake of it alike,
when the whole company forms a circle, by joining
their hands, and when the operator directs the person
who is at one extremity of the circle to hold the chain
which communicates with the coating, while he who is
at the other extremity of the circle touches the other
chain or wire. All the persons who form this circuit
being struck at the same time, and with the same de-
g?*ee of force, it is often very pleasant to see them all
start at the same moment, to hear them compare the|ir
seusatioBS, and observe the very difierent accounts they
give.
This experiment may be agreeably varied, if the ope-
rator, instead of making the company join hands, di-
rects them to tread on each other's toes, or lay their
hands on each other's heads. If, in the latter case,
the whole company should be struck to the ground^ as
it once happened, when Dr. Franklin gave the shock to
six very stout men, the inconvenience arising from it
will be very little; the company that was struck imme-
diately got up again, without knowing what had hap-
pened. This stroke was given with two jars, each o^
the measure of about six gallons, but not fully
charged.
292 AMUSEMENTS
Magical Explosions.
We have shown in a preceding experiment how gun-
powder may be fired by the intervention of spirits, but
there is another method, more simple and expeditious,
which we shall here describe. M5ake up gunpowder in the
form of a small cartridge, in each end of which put a
blunt wire, so that the ends within the cartridge may
be about half an inch distant from each other, then
joining the chain that comes from one side of the bat-
tery to one of the wires at the end of the cartridge,
bring the chain that comes from the other side of the
battei'y, to the wire at the other end, when the shock
will instantly pass through the powder, and set it on
fire.
By a similar method, fine brass or iron wire may be
melted ; for the explosion will pass from one chain to
the other, though the wire, which will be first red hot,
and then melt into round drops. A battery of 35 jars
has entirely destroyed fine brasd^ wire, of the 330th
part of an inch in diameter, so that no particle of it
could be found after the explosion. At the moment of
the stroke, a great number of sparks, like those from
a flint and steel, flew upward and laterally frona the
place where the wire was laid, and lost their light, in
the day, at the distance of about two or three inches.
A stroke from a common jar will easily strike a hole
through a thick cover of a book, or many folds of
paper, leaving a remarkable bur or prominence on both
sides, as if tl>e fire had darted both wayp from the
center.
The Prismatic Colour,
To the ends of each of the chains that come from
the battery, fix an iron wire, and between those wires
place a plate of tin, of about three inches square, and
polished on one side, in a perpendicular direction.-—
The wire next the polished side should be finely
pointed, and brought very near the surface of the
plate.
IN ELEcnacmr. 193
By repeating the explosions of the battery ^ there
iRrill first Appear a dusky red, about the edge of the
central spot; presently after, generally after four or
five strokes, there appears a circular space, visible only
in an oblique position to the light, ana looking like a
shade on the plate: this expands very little" during the
whole course of the explosions. After a few more
discharges, die second circular space is marked, by
another shade beyond the first of one-eighth or one-
tenth of an inch in width, which never changes its
appearance ^fter any number of explosions. AH the'
colours make their first appearance about the edge of
the circular spot ; more explosions make them expand
toward the extremity of the space first marked out ;
while others succeed in their place, till after 30 or .40
explosions, three distinct ring^ appear, each consisting
of all the colours in the prism or rainbow.
It makes no difference whether the electricity issue
from the pointed wire upon the plate, or from the plate
upon the pointed wire, the surface opposite the point
being marked exactly the same in both cases. The
points, themselves, from which the fire issues, or at
which it enters, are coloured for about half an inch to
a considerable degree, and the colours are repeated,
as on the plate.
The innermost, that is, the last formed colours, on
the plate, are always the most vivid, and those rings
ar^ also closer to each other than the rest. Those co-
lours may be brushed with a feather or the finger with-
out injury, but they are easily peeled off by the nail, or
any thing that is sharp.
' Th€ Artificial Spider.
Cut a piece of burnt cork, about the size of a pea,
into the form of the body of a spider; make its legs of
linen thread, and put a grain or two of lead into it. to
give it more weignt. Suspend it by a fine line of silk
between the electrified a^rch and an excited stick of
wax, and it will, like a clapper between two belk,
jump continually firom one body to the other, movinj^
194 AMUSEMENTS
its legs, at the same time, as if animated ; to the no
small suiprsie of those who are ijtxutcquainted with the
electric iimueiice*
The Artificial Earthquake,
In the middle of a large bason of water place around
wet board: this board represents the earth, and ^e
water the sea. On the board erect an edifice, com-
posed of several separate pieces, which may represent
a charcb, a castle, a palace, or, if you pkase, all of
them.
Then placing a wire that communicates with the two
chains of the battery, so that it inay pass over the
board and the surface of the water, upon making the
explosion, the water will became agitated, as in en
earthquake, and the board moving up and dawn, will
overturn the structures it supports; at the same time
that the cause of this commotion is totally concealed.
The Electrical Kite. .
Take a large thin silk handkerchief, and extend it,
by fastening the four comers to two slight strips of
cedar. The handkerchief thus prepared and accommo-
dated with a tail, loop, and string, will rise in the air
UKe a common paper kite. To the top of the upright
stick of the cross is to be fixed a pretty sharp pointed
;wire, rising a foot or more above the wood. To the
>end of the twine next the hand is to be tied a silk rib-
band, and where the twine and silk join, a key or tin
tube may be fastened.
This kite is to be raised when a thunder gust ap-
pears to be coming on, and as soon as the thunder
clouds come over the kite, the pointed wire will draw
the electricity from them, and the kite, with all the
twine, will be electrified, the loose filaments of the twine
will stand out every way, and be attracted by the finger.
When the rain has wetted the kite and twine, so that
it can conduct the electric fire freely, it will stream out
plentifully from the key, on the approach of a man's
IN ELECTRICITY. .1 35
knuckle. At this key a phial may be charged, and
from the electric fire thus obtained, spirits may be
kindled and all the other electric experiments per-
ibrmedy which are usually done by the help of a rubbed
glass or tube, and thereby the identity of the electriq
matter with that of lightning completely demon-
ctrated
The Candle lighted f>}i EUcfricity,
. Charge a sm^all coated phial^ whose knob is bent
outwaros so as to hang a little over the body of the
phial ; then wrap some loose cotton over the extremity
of a long brass pin or wire, so as to stick moderately
fast to its substance. Next roll this extremity of the
pin which isr wrapped up in cotton in some fine pow-
dered resin ; then apply the extremity of the pin or wire
to the external coating of the charged phial, and bring
as quickly as possible the other extremity that is wrap-
ped round with cotton to the knob: the powdered
resin takes fire, and communicates its fiame to the
cotton, and both together burn long enough to light a
candle. Dipping the cotton in oijiof turpentine will do
as well as if you use a larger sized jar.
Candle Bombs,
Procure some small glads bubbles, having a ne(^
about an inch long, with very slender bores, by means
of which a small quantity of water is to be introduced
into them, and the orifice afterwards closed up. This
stalk being put through the wick of a burning candle,
the fiame boils the water into a steam, and the glass
. is broken with a loud explosion.
Dancing Balls,
Take a common tumbler or glass jar, and having
placed a brass ball in one of the holes of the prime
conductor, set the machine in motion, and let the balls
touch the inside of the tumbler; while the ball touches
k2
196 AMUSEMENT9
only one point, no more of the surface of the glass will
be electrified, but by moving the tumblers about so as
to make the ball touch many points successively, all
these points will be electrified^ as will appear by turn-
ing down the tumbler over a number of piUi or corl:
balls placed on^a ts^ble. These balls will immediately
begin to fly about.
The Leaden Phial.
When a nail or a piece of thick brass wire, SfOi is
put into a small apothecary *s phial and electrified| re-
markable effects will follow; but the phial must be v^rj
dry or warm. Rub it once beforehand with ygur fin-
ger, on which put some pouoded chalk. If a little
mercury, or a few drops of spirit of wine, be put into
it, the etxperiment succeeds the better, As soon as this
phial and nail are removed firom the electrifying glass,
oc the prime conductor to which it has been ei^posed,
is taken away, it throws out a peqcil pf flaptie so long,
that with this burning n^achine in your h^nd, you may
take about sii^ty steps in wall^ing i^bout your room-
When it is electrified strongly, you may take it into
another room, and there fire spirits of wine with it.
If, while it is electrifying, y0u put your finger, or a
piece of gold which you hold in your hand, to the nail,
you receive a shock which stuns your arms and shouU
ders.
A tin tube, or a man placed upon electrics, is elec-
trified much stronger by tbis means than in the common
way. When you present this phial and nail to a tin
tube, fifteen feet long, nothing but experience can make
a person believe how strongly it is electrified. Two
thiti glasses have been broken by the shock of it. It
appears extraordinary, that when this phial and nail are
in contact with either conducting or non-conductiqg
matter, the strong shock does not follovf,
IN ELECTRICmr. 197
Rosin ignited hy Electricity.
Wrap some cotton wool, containing as much pow-
dered rosin as it ^ill hbld, about one of the knobs of a
dischargihj^ rod. Theh having charged a Ley den jar,
apply the naked knob of the t-od to the external coating,
and the knob enveloped by the cotton to the ball of the
wire. The act of discharging the jar will set fire to
the rosin.
A piece of phosphorus or camphor wrapped in cotton
wool, and used in the same way, will be much more
easily inflamed.
Spirits ignited by Electricity,
tlang St Sttiail ball \nih a stem to the pHme con-
ductor, sb that the ball may project below the <ionduc-
tor. Then Watm a little ardent spirit, by h'dlding it a
short time oVeV a ^ndle in a metallic spoony hold the
spoon about an inch below the ball, and )iet the ma-
tmne in motion. A spark Will soon issue from the
ball, and set fire to the spirits.
- This experiment may be varied diJBTerent Ways, and
may be rendered very agfeeable to a company of spec-
tators* A person, for instance, sfanditigupon an elec-
tric stool, and communicating with the prime conductor,
may hold the spoon with tile spirits in his hand, and
another person, standing upon the floor, may set the
spirits On fire, by bringing his finger within a small
distance of it. Instead of hi« finger, he may fire the
Spirits with a piece of ice ; When the experiment will
seem much more surprising. If the spoon be held by
the person standing upon the floor, and (he insulated
person bring some conducting substance over the sur-
face of the spirit, the experiment succeeds as well.
Electrified Air.
Fix two or three pointed needles into the prime con-
ductor of an electrical machine, and set the glass in
motion so as to keep the prime conductor ekctxv^%^
k3
198 AMU8CM£HT»
for several minutes. If now, an electrometer be brought
within the air that is contiguaus to the prime con-
ductor, it will exhibit signd of electricity, aod this. air.
will continue electriBed for some time, even after the
machine has been removed into another room. The
air, in this case is electrified positively ; it may be ne-
gatively electrified by fixing the needles in the nega-*
tive conductor while insulated, and making a commu*
nication between the prime conductor and the table, by
means of a chain or other conducting substance. .
The air of a room may be electrified in another way.
Charge a large jar, and insulate it; then connect two
or more sharp pointed wires or needles^ with the knob
of the jar, and connect the outside coating of the jar
with the table. If the jar be charged ^itively, the
air of the room will soon become positively electrified
likewise ; but if the jar be diarged negatively, the elec*
tricity communicated by it to the air, will become also
negative. A charged jar being held in one iiand, an4
the flame of an msulated candle held in. the other
being brought near the knob of the jar^ ._ will also pro-
duce the same efiect.
To Spin Sealing-wax into Threads by Ekc^citjf,
Stick a small piece of sealing-wax on the end of 3
wire, and set fire to it. Then put an electrical machine
in motion, and present the wax just blown at the dis-
tance of some inches from the prime conductor. A
Qumber of extremely fine filaments will immediately
dart from the sealing-wax to the conductor, on which
they will be condensed into a kind of net-work, re-
sembling wool.
If the wire with the sealing-wax be struck into one
of the holes of the conductor, and a piece of paper be
presented at a moderate distance from the wax, just
after it has been ignited, on setting the machine in mo-
tion, a net-work of wax will be formed on the paper.
The same effect, but in a slighter degree, will be pro-
duced, if the paper be briskly rubbed with a piece of
IN tiLECTRICITY. 199
eta«tic ^m, and the melting sealing-wax be pretty near
the paper immediately after rubbing.
If the paper thus painted, as it were, with sealing'*
wax, be gently warmed by holding the back of it to the
fire, the wax will adhere to it, and the result of the
experiment will thus be rendered permanent.
The Electrified Camphor.
A beautiful experiment of the same nature is made
with camphor. A spoon holding a piece of lighted
camphor is made to communicate with an electrified
body, as the prime conductor of a machine; while Ae
cendnctor continues electrified by keeping the maclmie
in motion, the camphor will throw out ramificattonSy
and appear to shoot like a vegetable.
ELECTRICAL AMUSEMENTS IN THE DARK
CHAMBER.
To exhibit a great number of pleasing and surprising
amusements in the dark, as well as m the light, is
the peculiar property of electricity : for though there
are many beautiful experiments performed in the camera
obscura, it is still by the aid of the sun's rays, or those
of a candle or lamp: whereas, the electric apparatus
contains within itself the particles of the fire by which
these amusements are performed.
The Fiery Shower.
On the plate put a number of any kind of seeds,
grains of sand, or brass dust. The conductor being
strongly electrified, those light particles will be at-
tracted and repelled by the plate suspended from the
conductor, with amasing rapidity, so as to exhibit a
perfect fiery shower.
Another way is, by a sponge that has been soaked
in water. When this sponge is first hung to the con-
ductor, the water will drop from it very slowly; but
when it is electrified, the orops will fall very fast, and
K 4
300 AMU8EMEKTS
appear like small globes of firev llluminatiDg the basiir
into which they falL
The Miraculous luminaries.
To perforin this amusement it is necessary to be pro^
vided with a quantity of the following phosphorus.. —
Calcine common oyster^shells by burning them in the
fire about half an hour ; then beat them into powder,
of the clearest of which take three parts, and of flow-
ers of sulphur one part, and put the mixture into a
crucible about an inch and a half deep. Let it burn
in a strong open fire for a fiilFhour; when cool, ttura
it out, and- break it into several pieces, and taking those
pieces into & dark place, scrape oflf the brightest parts
for use, which, when good, wilf be a white powder.
Then take a circufar board of three or four feet dia*
meter, on the center of which draw the figure of the
half moon, of three or four inches diameter, and round
it, at different distances, draw a number of stars, of
different magnitudes. On each of these figures fix the
phosphorus just mentioned, to the thickness of about a
quarter of an inch. The board beii^ thus prepared,
you must have ready a number of charged jars or phial's,
and by discharging one of themy at the distance of
about an inch, over each figure, it will become illumi-
nated. The light of the crescent will be so strong at
first, that you may distinguish by it the figures on the
dial of a watch. Rouncf the board let there be placed
a rim or hoop, and over that, at a sufficient distance
from the figures draw a curtain.
The board thus prepared is to be brought into the
darkened room, and placed, by hooks, against the ceil-
ing. The curtain is then to be drawn back, and the
moon and stars will then appear as emerging from be-
hind a cloud, and will continue to shine for half an
hour: the light, however, growing continually more
faint.
IN ]SL£CTRictry. 30L
The Globular Fires,
r
Let the room, and all the parts of the apparatus, be
made very dry, and let the globe be strongly excited^
so that the electricity may be very vigorous; the fire
will then be seen to dart from the cushion toward the
wire of the conductor. Sometimes these lucid rays
(which are in part visible in day-light) will make the
circuit of half the globe, and reach the wires ; and
they will frequently come in a considerable number, at
the same time, from different parts of the cushion, and
reach within an inch or two of the wires, the noise at-
tending this beautiful phenomenon exactly resembles
the crackling of bay^eaves in the fire. These lucid
arches have frequently radiant points, often four or five
in different parts ot the same arch. Tliese radiant
points are intensely bright^ and appear very beautiful.
It is peculiarly pleasing to observe the circles of fire
rise from those parts of the cushion, where the amal-
gam or moisture has been put, or which have been
lately scraped. Single points on the rubber will then
appear intensely bright, and for a long time together
will seem to pour out; continual torrents of flame. If
one part of the rubber be pressed closer than another,
the circles will issue from that part more frequently
than from any other.
When the conductor is taken quite away, circles of
fire will appear on both sides the rubber, which will
sometimes meet and completely encircle the globe. If
in that state, a fmger be brought within half an inch of
the globe, it is sure to be struck very smartly; and
there will often be a complete arch of fire from it to the
rubber, though it be almost quite round the globe.
If air the air be exhausted from the. globe, the elec-
tricity will be found to act wholly within it, where it
will appear in the form of a cloud or flame of teddish
or purple-coloured light, filling the whole interior space
of the globe. When this amusement is finished, the
globe and rubber must be taken away, that they mciy
not incommode the apparatus of the K>llowing experi-
ments.
k5
• - • ••
The Illuminated Vacuum*
' Take a ttU receiver that is very dry, and through the
top of it fix, with cement, a wire, hot rery acutely
Knted. Then exhaust the receiver, and present the
)b of the wire to the conductor, and every spark will
pass through the vacuum, in a broad stream of light,
visible thirough the whole length of the receiver, how
tall soever it be. This stream often divides itself into
a variety of beautiful rivulets, which are continually
changing their course, uniting and dividing again in a
most pleasing manner. If a jar be discharged through
thi« vacuum, it gives the appearance of a Very dense
body of fire, darting directly through the center of the
vacuum, without ever touching the sides; whereas,
when a single spark passes through, it generally goes
Inore or less to the side, and a finger put outside cf the
glass, will draw it wherever a person plesuse. If the
vessel be grasped by both hands, every spark is felt,
like the pulsation of a large artery, and all the fire
makes towards the hands. This pulsation' is felt at
some distance from the receiver, and a light is seen
between the hands and the glass.
* All this while the pointed wire is supposed to be
electrified positively ; if it be electrified negatively the
appearance is remarkably different. Instead of streams
of fire, nothing is seen but one uniform luminous ap-
pearance, Kke a white cloud, or the milky way in a
clear star light night. It seldom reaches the whole
length of the vessel, but generally appears only at the
end of the wire, like a lucid ball.
If in the neck of a tall-receiver a small phial be in-
serted, so that the external surface of the glass may
Ije exposed to the vacuum, it ydll produce a very beau-
tiful appearance. The phial must be coated on the
inside, and while it is charging, at every spark taken
fi^om the conductor into the inside, a fiash of light is
seen to dart, at the same time from every part of the
external surface of the phial, so as to quite fill' the re-
ceiver. Upon makhig the discharge, the light is seen
to return in a much closer body, the whole coming out
at once.
IN EIECTRICITV. W8
The Luminous Cylinder*
Provide a glass cylinder three feet long and three
inches diameter : near the bottom of it fix a bniuM
plate, and have another brass plate so contrived that
you may let it down the cyUnder, and bring it as tie^
the first plate as you desire. Let this cyUnder be ex-
hausted and insulated, and when the upper part is
electrified, the electric part will pass from one plate to
the other, when they are at the greatest distance from
«ach other the cylinaer will admit* The brass plate at
the bottom of the cylinder will moreover be as strongly
electrified, as if it was connected by a wire with the
prime conductor.
The electric matter in Its passage thnough this va-
cuum is said to produce a delightful spectacle; not
making, as in the open air, small brushes or pencils 6f
rays, an inch or two in length, but coruscations of the
whole length of the tube, and of a bright silver hue.
These do not immediately diverge as in the open air,
but fVequently form abase that is apparently flat, divid-
ing themselves into less and less ramifications, and
very much resemble the most lively coruscations of the
aurora borealis.
The Magical Constellations.
As the moon and stars in the zenith will become dull
during the time of pefforming the preceding amuse-
ments, it will be proper to draw the curtain gently be-
fore them, that it may seem as if a cloud came slowly
over them; and then the operator riftiy, by his magical
power^ light up other constellations. In order to which,
he must provide a large board, on which let him mai4c
ttie stars that are in two or more constellations, which
ate contiguous and visible in the northern hemisphere,
as Taurus, Gemini, &c; v
To represent these stjurs, let th^re be a hole on each
aide of Um spot that k marked fbr a star, about a
quarter of an mch distant firom each other, and let the
k6/
.*'
234 AmmEMBNTS
extremities of two wires, neatly rounded, come through
these holes, and be brought near together, exactly over
the tnaric. These wires should be of different sizes,
that they may the better represent the different mag-
nitudes of the stars.
' The other ends of the wires must be so disposed, tliat
they may all receive a spark from the conductor at the
same time, and the stars will then be all luminous at
the same instant These stars are not eranescent, like
those made by the phosphorus-, but will continue with
equal splendour, as* long as the motion of the wheel is
continued. After the same manner any cipher^ or the
outlines of a drawing may be exhibited.
The Aurora Borealis^
Make a Torricellian iFaeuum in a glass tube, about
three feet long, .and seal it hermeticaliy i"^ it will then
be always ready for use. Let one end of this tube be
held in the hand, and the other be applied to the con-
ductor, and immediately the whole tube will be illumi-
nated from end to end : and when taken from the
conductor will continue luminous, without interruption,
for a considerable time, very often above a quarter of
an hour. If, after this, it be drawn through the hand
either way, the light will be uncommonly intense, and
without the least interruption, from one hand to the
Other, even to its whole length. After this operation,
which discharges it in a great measure, it will still
flash at intervals, though it be held only at one extre-
mity, and quite still; but if it be grasped by the otlier
hand at the same time, in a different place, strong
flashes of light wiH hardly ever fail to dart from one end
to the other; and this will continue 24 hours, and
perhaps much longer, without fresh excitation. Small
and long glass tub^ exhausted of air, and bent in many
irregular crooks and angles, will, when properly ^lec^i-
fled, beautifully represent flashes of lightning.
* The Toricettian vaciiam it nisde by iilfing a tube witli pare mer-
cury, and then iiiTerthig il, In the tame manner ai in nakiag a ftaro'
meter ; for as the mercury raifl out, all the tpace above will oe a true
Vacuum. A glass Is heroiatkally sealed by holdiilg the end of it ia
the flame of a candle, till it b ready to melt, and then twbtin^ it to-
gether with a pair of pincers.
IN ELECTfticiry. S05
Tht Ciradating Lamps,
After keeping the. company tlius long in tbe dark, it
will be proper to illuminate the room before you dis-
miss them. In order to which, introduce the circulating
wheel described in page 186. To the upper axis of which
let there be fixed a number of radii, made of baked
wood, at the end of each of which must hang a small
globular lamp, filled with spirits; and let that of eacfi
lamp be ringed with a different colour. The wheel,
havmg previously acquired its greatest velocity, is 1o
be placed on the table, and a chain, depending from
the branch, is to dip into each lamp as it passes by ;
so that alt of them will become illuminated in a very
short time. These lamps will not only enlighten the
room, but by their variegated colours, and continual
revolution, afford a very pleasing plienomenoBr
20 1 AMUSEMENTS
MAGNETISM.
DEFINITIONS.
1. MAGNETISM la the science that explains the
several properties of the attractive and repellent powers
in the magnet or loadstone.
if 2. The magn^et is a rich, heavy, iron ore of a hard
substance, a dusky ^rey colour, with some mixtures of
a reddish brown, and sparkling when broke.
3. The magnetic virtue is called the third species of
attraction ; gravity being the first, and electricity the
seconds
4. The two ends of a magnet, when it is properly
formed, are called its poles ; and when it is placed on
a pivot, in just equilibrium, one end will turn toward
the north, and is called its north pole, and the other
end the south pole.
6. When the two poles of a magnet jare surrounded
with plates of steel, it is said to be armed.
6. if the end of a small iron bar be rubbed against
one of the poles of a magnet, it is said to be touched,
and is then called an artificial magnet
7. If such a magnet be supported on a pivot, ii is
called a magnetic needle ; one end of it turning toward
the north, and the other toward the south.
8. The difference between the position of the needle,
and the exact points of north and south, is called its
declination.
9. A needle which is touched will Incline toward the
earth, Imd that is called its incUnatiofi or dippiii;g^.
IN MAGNETISM. CO 7
APin)R18MS.
1 . The magnetic attractioo is produced by effluvia
emitted by the magnet, and passing iram one pole to
tlie other.
2. One pole of a magnet will attract iron, and the
other repel it, but no other body.
3. The magnet atiracts iron as well in racuo as in
the air.
4. The magnetic attraction will be continued through
several pieces of iron placed contiguous to each other.
5. The magnetic effluvia pervades all bodies.
6. The magnetic attraction extends to a considera-
ble distance.
7. The north pole of one magnet will attract the
south pole of another: and the similar poles will repel
each other,
8. The end of a needle touched by the north pole
of a magnet will turn south, and that touched by the
south pole will turn north.
9. The declination of the magnetic needle is different
in different parts of the earth, and in the same part at
different times.
10. The inclination of the needle is not always the
same in different places, nor in the same pUoe at dif-
ferent times.
1 1 . The strength of natural magnets differs in those
of different magnitudes, but not in proportion to their
magnitudes.
12. The strength of a natural magnet is considera^
bly increased by its being armed.
1 3. Iron acquires a magnetic power by^ beinf oon-
iinually rubbed in the same direction.
14. Iron bars become magnetic by standing^ n haxg
time nearly upright.
15. The magnetic virtue may be communicated by
electricity.
16. A strong blow at one end of a magnetic bar will
give it a magnetic power.
17» Ftr9 totaUjf 4e»My8 tbepowec of magMiiy. M
well natural ai$ artificial.
t08 AMUSEMEKTtf
The Magnetic tVand.
Bore a hole, three tenths of an inch d'tametef , through
a round stick of wood; or. get a hollow cane about
eight inches long, and half an inch thick. Pi^ride a
small steel rod, and let it be very strongly impregnated
with a good magnet; this rod is to be put in the hole
you have bored through the wand^ and closed at each
end by two small ends of ivory that screw on, different
in their shapes, that you may better distinguish the
poles of the magnetic bar.
When you present the north pole of this wand to the
south pole of a magnetic needle, suspended on a pivot,
or to a light body swimming on the surface of the
water (in which you have placed a magnetic bar,) that
body will approach the wand^ and present that end
which contains the south end of the wand, to the north
or south end of the needle, it will recede from it.
The Mifsterious Watch*
ITou. destr^ aify person to letid you his watch, and
ask him if it will go when laid on Uie table. He will,
no doubt, say it will ; in which case, you place it over
the end of the magnet, and it will presently stop. You
then mark the precise spot where you placed the watcK
and moving the point of the magnet, you give the
watch to- another person, and desire him to make the
experiment ; in which he not succeeding, you give it
to a third (at the same time replacing the magnet) and
he will immediately perform it«
This experiment cannot be effected^ unless you use
1^ very stronffly impregnated magnetic bar, (which may
be purchased at the opticians*,) and the balance of the
watch must be of steel, which may be easily ascertained
by previously opening it^ and looking at the works.
The Magnetic Diah
Procure a circle of wood or ivory, of about dve or
its inches diameter, which must turn quite free 0u a
In magnetism. 209
fttand with a circular border ; on the ivory or wood cir-;
cfe fix a pasteboard, on which you place, in proper di-
visions, the hours, as on a dial. There must be a small
groove in the circular frame to deceive the pasteboard
circle, and obsefve,- that the dial must be fnade t6 turn
so free, that it may go round without moving the circu^
lar border it which it is placed.
Between the pasteboard circle and the bottom of the
frame, place a small artificial magnet, that has a hole
in its middle. On the outside of the frame, place a
small pin, which serves to shew when the magnetic
needle is to stop. This needle must turn quite free
on its pivot, and its two sides should be in exact equi-
librio.
Then provide a small bag, with five of six divisions^,
like a lady's work bag, but smaller, hi oiie of these '
divisions put small square pieces! 6f pastebbai^d, om-
vhich are written the numbers from 1 to 12. tn eacfi
of the other divisions put twelve or more similar pieces,
observing that all the pieces in each division must be
marked with the same number. The needle beinr
placed upon its pivot, and turned quickly about, it wiU
necessarily stop at that point where the north end of
the magnetic bar is placed, and which you previously
know, by the situation of the small pin in the circular
border.
Yon then present to any person that division of the
ba^ which contains the several pieces, on which if
written the number opposite to the north end of th«>
bar, and tell him to draw any one he pleases. Them
placing the needle on the pivot, you turn it quickly
about, and it must necessarily stop at that particuUor
number.
The Magnetic Cards,
Draw a pasteboard circle ; you then provide your-
self with two needles, similar to those used in the fore-;
going experiment, (which you must distinguish by some
private mark) with their opposite points touched with
the magnet. When you place that needle^ whose
210 AM0lfiM£NT8
pointed end is touched, on the pivot described in thd .
centre of the circle, it will stop on one of the four pips^
against which you have placed the pin in the frame ;
then take that needle on, and placing the other^ it will
stop at the opposite point.
Having matters thus arranged, desire a person to
draw a card from a piquet pack^ ofifering that card
against which you have placed the pin of the dial,
vmich you may easily do, by having a card a lltde
longer than the rest. If he should not draw it the first
time, as he probably may not, you must make some
excuse for shuffling them again; such as letting the
dsrds fall, as if by accident, or some other manoeuvre,
till be fixes on the card. You then tell him to keep
it dose, and not let it be seen. Then eive him one of
the two needles, and desire him to place it on the
pivot, and turn it round, when it will stop at the coloor
of the card he chose ; then taking that needle off, and
exchanging it unperceived for the other, give it tp a
second person, telling him to do the same, and it will
stop at the name of Uie identical card the first person
The Communicative Crown.
Take a crown piece, and bore a hole in the side of
it ; in which place a piece of wire, or a large needle
well polished, and strongly touched with a magnet.—
Then close the hole with a small piece of pewter, that
it may not be perceived. Now the needle in the mag-
netic perspective before described, when it is brought
near to this piece of money, will fix itself in a direction
correspondent to the wire or needle in that piece.
Desire any person to lend you a crown piece, which
you dexterously change for one that you have pre-
pared as above. Then give the latter piece to another
person, and leave him at liberty either to put 'it pri*
vately in a snuff-box or not. He is then to place the
box on a table, and you are to tell him, by means of
your |;laBS, if the crown is, or is not in the box. Then
bringing your perspective close to the box, you will
IN MAGNETISM. ^11
know, by the motion of the needle, whether it bei there
or not ; for as the needle in the perspective will always
keep to the north of itself, if you do not perceive it has
any motion, you conclude the crown is not in the boit.
It may happen, however, that the wire in the crown'
may be placed to the north, in which case you will l>e
deceived. Therefore, to be sure of success, when yoil
find the needle in the perspective remain stationary,
you make some pretence to desire the person to move
the box into another position, by which you will cet-'
tainly know if the crown piece be there or not.
You must remember that the needle in the perspM«*
live must here be very sensible, as the wire in the
crown oamiot possibly have any great attractive force.
The Magnetic Table,
Under the top of a common table place a magnet
that turns on a pivot, and fix a board under it, that
nothing may appear. There may also be a drawer
under the table, which you pull out to show that there
is nothing concealed. At one end of the table there
must be a pin that communicates with the magnet, and
by which it may be placed in different positions : this
pm must be so placed as not to be visible by the spec-
tators. Strew some steel filings, or very small nails,
over that part of the table where the magnet is. Then
ask any one to lend you a knife, or a key, which will
then attract part of the nails or filings, in the same
manner as the iron attracts the needle, in the note to
the twelfth aphorism. Then placing your hand, in a
careless manner, on the pin at the end of the table,
you alter the position of tlie magnet : and giving the
key to any person you desire him to make the experi-
ment, which he will then not be able to perform. You
then give the key to another person, at the same time
placing the magnet, by means of the pin,, in the first
position, when that person will immediately perform the
experiment.
The Ineomprehendble Card,
insert in the middle of a card, amd parallel to it»
two longest sides, part of a teatch spring, as thin at
possible, and strongly impregnated : let it be so con-
cealed as not to afford the least suspicion. This card
should be a little longer than the others of the pack in
which it is placed.
Offer any one to draw a card ont of the pack, and
present the long card dexterously to his hand. You
then give him all the cards, and leave him to replace
that card in the pack or not. He is then to lay the
pack on the table, and by applying your magnetic per-*
spective, you will discover whether the card be there or
not
If the person should not draw that card, you must be
ready with some other experiment, to prevent suspicioa
of having fieuled in your aesign.
SIS
PNEUMATICS.
PEFJNITI0N9.
1. THE atmosphere is that body of air which erery
where surrounds the earth.
•2. The air pump is a machine contrived to produce
a vacuum, by exhausting the air out of the vessel called
a receiver.
3. The condenser is an instrument generally In form
of a syringe, to force a greater quantity of air into any
vessel than it naturally contains.
4. The anometer is an instrument that measures the
strength of the wind.
5. The hygrometer is contrived to show the different
degrees of moisture in the atmosphere at different times.
6.^ The thermometer measures the degrees of heat
and cold of the air, and of other bodies.
7. The barometer shows the diflferent weight of the
air at difierent times.
APIIOEISMS.
1. The air is an elastic, ponderating, compressible
and expansible fluid, that is sensible only to the touch.
2. The elasticity of the air is increased by heat and
decreased by cold.
3. The weight of the air is so small as not to be per-
ceived but in large quantities.
4. The rarefaction and condensation of the ait, are
indefinite.
5. Though air is greatly condensible by cold, it can-
not be congealed.
6. Air is necessary to animal existence.
dl4 AMU3BHENT6
7. Adust air, that is, such as has passed through the
fire or a heated tube, will not support shiimal life.
8. Air is conCaiaod in almost ^. bodies, and may
be produced from them.
9. Sound is communicated by the air.
10. The atmosphere is of different densities at dif-
ferent heights, and is most dense near the earth,
Ih The height of the atmosphere does not exceed
$0 miles.
12. Wind is nothing but a current of air.
13. The velocity of the wind is from 1 to 60 miles in
•an hour.
To describe the numerous apparatus necessary for
.etperimenting on air, among which the Air Pump^ the
Animometer, and Hygrometers, are the most conspi-
cuously useful, would occupy more space than the li-
mits of our small volume can allow. We shall theie-
fore refer the reader to works professedly devoted to
reseftfches in Natural Philosophy, for descriptions and
illustrations of these instruments, and proceed' to enu-
merate a few amusing experiments in this branch of
science. ^
%.
The Bottle broke bj/ Air.
Take a bottle that is square, not round or cylindri-
cal ; and if it be small, the glass must be thin. Put
the mouth of this bottle over the hole in the place of
the air-pump, and exhaust the air. By this means the
bottle will be made to sustain the weight of the exter-
nal ak as long as it is able, but at last it will be sud-
denly burst into very small parts.
The same effect may be produced by the spring of
the air, in the follov^ing manner. Seal the mouth of
a bottle so close that not the least air can come out,
and place it in the receiver; then as the air is drawn
off from its surface, the spring of the included air will
act against the sides of the bottle, and will continually
increase as the air in the receiver becomes more, raid-
fied till at last it burst the bottle in piepe's*
,IN PNUEMATIC8. 215
A similar effect is produced by laying a plate of glass
on the top of an open receiver, and exhausting the
air ; for then the weight of the external air will press
upon the glass and iH'eak it in pieces. In like man-
ner, if a person lay his hand upon an open receiver,
and the «lr be exhausted, his hand will be fixed to the
receiver : for if the aperture of the receiver be four
square inches, the weight on his hand wiU be equal to
60 pound. This experiment will be attended with some
pain in the person's hand.
The Brass Hemisphere.
Take two hemispheres of about four inches diameter,
and whose circumferences exactly fit each other. Now,
when they are placed together, aud the air is exhausted
from their cavities, the internal spring taken away,
they will be pressed by a column of air equal to their
surfaces, that is,. twelve square inches and a half, which,
multiplied by 15 pounds, the weight of the air on every
inch, the sum will be 187 pounds and a half.
Therefore, give these hemispheres to any two' per-
sons, after they have seen them put together, and that
they are not in any manner joined to each other, and
desire them to pull the hemispheres asunder; to effect
which they must, between them, exert a force equal to
the above number of pounds.
JValer boiled h^ Air.
Take water that is made as warm as ypu can well
bear to put your hand in it, but that has not boiled,
and putting it under the receiver, exhaust the air.r— .
Buboles of air will soon be seen to rise, at first y&cj
small, but presently become larger, and will be at last
so great, and rise with such rapidity, as to give the
water all the appearance of a violent boiling. This
agitation of the water will continue tiU the air is again
. let into, the receiver, when it will immediately ce^M,
and the water become quite motionless.
'dl6 ABfUSEMEim
The JErial Bubbles.
Take a piece of iron, brass, stone, or any other heavy
•ubtftance, and putting it in a large glass with water,
place it in the receiver. The air being exhausted, the
spring of that which is in the pores of the solid body,
by expanding the particles, will make them lise on' its
surface in numberless globules, which resembling the
pearly drops of dew on the tops of the grass, afford a
rery pleasing appearance. On letting the air into the
receiver, all these eerial forms immediately disappeair*
The Floating Stone,
To a piece of cork tie a small stone, that will just
sink it, and putting it in the vessel of water, place it
under the receiver. Then exhausting the receiver, the
i>ubbles of air which expand from its pores, and ad-
here to its surface, will render it, together with the
stone, lighter than water, and consequently they will
rise to the surface and float.
The withered Fruit restored.
Take a shrivelled apple, and placing it under the
receiver, exhaust the air. The apple will immediately
be plumped up, and look as fair as when first gathered.
For the pressure of the external air being takex^ off,
the expansion of that contained within the skin of the
apple will extend it to the utmost, so as sometimes to
make it burst. This restoration, however, is merely
apparent, for the air is no sooner let into the receiver
again, than the apple returns to its former withered
state^
The Vegetable Air Bubbles.
Put a small branch of a tree with its leaves, or part
of a small plant, in a vessel of water, and placing the
vessel in the receiver, exhaust the air. When the pres-
IN PNEUMATICS. 217
»ure of the external air is taken ofF, the spring of that
contained in the air vessels of the plant, by expanding
the particles, will make them rise from the orifices of
all the vessels, for a long time together, and produce a
beautiful appearance. This experiment shows how
great a quantity of air is contained in every vegetable
substance.
The Mercurial Rod.
Take a piece of stick, cut it even at each end ^ith
A penknife, and immerse it in a vessel of mercury.
When the air is pumped out of the receiver, it will, at
the same time, come out of the' pores of the wood,
through the mercury, as wilt be visible at each end of
the stick. When the air is again let into the receiver,
it falls on the surface of the mercury, and forces it
into the pores of the wood, to possess the place of the
air. *
When the rod is taken out and weighed, it is found
to be several times heavier than before, and has
changed its colour, being now all over cf a bluish hue.
If this stick be cut transversely, the quicksilver will b%
seen to glitter in every part of it,
T//C Mystical Bell.
Fix a small bell to the wire that goes throtigh th^
top of the receiver, and shaking it by that wire it will
be distinctly heard, while the air is in the receiver. Aai,
the air is exhausted, the ringing becomes gradually
weaker, and at last, how much soever the bell be shook,
the least sound cannot be heard. But when the a^
begins to enter again into the receiver, the sound bd-
' comes presently audible. This experimeot proves thfCt
air is the medium of sound.
318 AMVaJEMESTfi
Feathers heavier than Lead,
At one end of a fine balance hang a piece of lead, and
at the other as many feathers as will keep it in equi-
librio. Then place the balance under the receiver. As
soon as the air begins to be exhausted, the equilibrium
will begin to be destroyed, and when all the air is ex-
hausted, the feathers will descend and the lead mount
up.
The cause of this phenomenon is plainly deducible
from the laws of hydrostatics ; for when both bodie«
are weighed in air, each looses the weight of an equal
bulk of air ; consequently the feathers will lose a greater
weight than the lead ; but when the air is taken away,
the weight that is restored to the feathers being greater
than that restored to the lead, the former will neces-
sarily preponderate.
The self-moving Wheel,
Take a circle of tin, about ten inches diameter, or
of any other dimension that will go into the receiver,
and to its circumference fix a number of tin vanes, each
about an inch square. Let this wheel be placed, be-
tween two upright pieces, on an axis whose extremities
are quite small, so that the wheel may turn, in a verti-
cal position, with the least force possible. Place the
wheel and axis in the receiver and exhaust the air. —
Let there be a small pipe with a cock ; one end of this
pipe is to be on the outside of the top of the receiver,
and the other end to come directly over the vanes of
the wheel.
When the air is exhausted from the receiver, open
the cock just mentioned. A current of air will rush
against the vanes of the wheel, and put it in motion;
and the velocity of its motion will increase till the re-
ceiver is again replete with air.
If the pump be kept continually working, after th«
air is exhausted, the motion of this wheel may be re-
garded not only as spontaneous, but perpetual.
IN ^PNEUMATICS. 319
The animated Figures,
Provide nine, twelve, or any number you please, of
hollow cylinders, about nine inches long, and one and
a half, or two inches diameter. Let the bottom of each
of these cylinders be closed, except a small hole ; and
in each of them place a piston, like that in a syringe.
At the bottom of each piston let there be a worm,
spring, and over it the figure of a man, woman, or what
else you please. These figures should be all different,
and in different attitudes, and of such a size that they
may completely enter the cylinders.
Place all the cylinders in a circular frame of wood,
and having pushed each piston down to the bottom of
the cylinder, and stopped the holes at bottom, draw it
lip again to what height you think proper, and there
will then be a vacuum under each piston. Then place
the frame in the receiver, and exhaust the air.
When the weight of the external air begins to be
taken off, the force of the spring that is at the bottom
of each piston being greater than its friction, and the
weight of the figure placed over it, they will gradually
rise up, and present themselves in their proper attitudes.
When the air is again let into the receiver, they will,
in like manner, retire to their separate apartments.
If the arms and legs of the figures be inflated with
a due quantity of air, when the pressure of that in the
receiver is taken off, they \^ill oe extended, and may
be made to assume any attitude : and when the air is
again let into the receiver, they will resume their former
positions.
The Artificial Halo.
Place a candle on one side of a receiver, and let the
spectator place himself at some distance from the other
side. As soon as the air begins to be exhausted, and
becomes attenuated, and charged with vapours to a
proper degree, the light of the candle will be refracted
through that medium in circles of various colours, that
lively resemble those seen about the moon in a haiy
night.
L 2
220 AMUSEMENTS
The Mercurial Shower.
Cement a piece of wood into the lower part of the
neck of an open receiver, and pour mercury over it —
After a few strokes of the pump, the pressure of the
air on the mercury will force it through the pores of
the wood in form of a beautiful shower : which, if the
receiver be clear and the weather be dry, will appear
luminous in a dark chamber.
The Fountain in Vacuo,
Take a tall glass tube, hermetically sealed at the
top and at bottom, by means of a brass cap, screwed
on to a stop cock, and that to the plate of the pump.
When all the air is exhausted the cock is turned, the
tnbe is taken off the plate and immersed in a bason of
mercury or water ; then the cock being again turned,'
the fluid, by the pressure of the air, will play up in the
tube, in form of a fountain, and afford a very pleasing
appearance.
The Cemented Bladder.
Tie the neck of a bladder to a stop cock, which is
to be screwed to the plate of the pump, and the air
exhausted from the bladder ; then turn the stop cock
to prevent the re-entrance of the air, and unscrew the
whole from the pump. The bladder will be transformed
into two flat skins, so closely applied together, that the
strongest man cannot raise them half an inch from each,
other; for an ordinary sized bladder, of six inches
across the widest part, will have one side pressed upou
the other with a force equal to 396 pounds weighty
Cork heavier than Lead.
Let a large piece of cork be pendent from one end of
a balance beam, and a small piece of lead from the
other; the lead should rather preponderate. If this
apparatus is placed under a receiver on the pump,
IN PNEUMATICS. 231
you will find that when the air is exhausted, the lead,
which seemed the heaviest body, will ascend, and the
cork outweigh the lead. Restore the air, and the effect
will cease. This phenomenon is only on account of the
difference of the size in the two objects. The lead,
which owes its heaviness to the operation of the air,
yields to a lighter because a larger substance when
deprived of its assistance.
The animated Bacchus^
Construct a figure of Bacchus, seated on a cask ; let
his belly be formed by a bladder, and let a tube pro-
ceed from his mouth to the cask. Fill this tube with
coloured water or wine, then place the whole under the
receiver. Exhaust the air, and the liquor will be thrown
up into his mouth. While he is drinking his belly will
expand.
The Artificial Ballooju
Take a bladder containing only a small quantity of
air, and a piece of lead to it, sufficient to sink it, if
immersed in water. Put this apparatus into a jar of
water, and place the whole under a receiver. Thea
exhaust the air, and the bladder will expand, become
a balloon lighter than the fiuid in which it fioats, and
ascend, carrying the weight with it.
Experiment icith a Viper.
Many natural philosophers, in their eagerness to dis-
play the powers of science, have overlooked one of the
first duties of life — humanity; and, with this view, have
tortured and killed many harmless animals, to exem-
plify the amazuig effects of the air-pump. We will not
stain the pages of this work by recommending any such
fpecies of cruelty, which, in many instances, can merely
gratify curiosity ; however, as many of our readers
might like to read the effect on animals, we extract
from the learned Boyle^ an account of his experiment
oo a viper.
l3
823 AMUftH^ENTS
He took a newly-caught viper, and shutting it up in
a small receiver, extracted the air. At first, upon the
air's being drawn away, it began to swell : a short time
after it gaped and opened its jaws; it then resumed its
former lankness, and began to move up and down
within the receiver, as if to seek for air. After a while
it foamed a little, leaving the foam sticking to the in-
side of the glass ; soon after, the body and neck be-
came prodigiously swelled, and a blister appears on its
back. Within an hour and a half from the time tho
receiver was exhausted, the distended viper moved, be-
ing yet alive, though its jaws remained quite stretched;
its black tongue reached beyond the mouth, which had
also become black in the inside. In this situation it
continued for three hours; but on the air's being re-
admitted, the viper s mouth was presently closed, and
soon after opened again; and these motions continued
some time, as if there were still some remains of life.
It is thus with animals of every kind ; even minute
microscopical insects cannot live without air.
Experiments with Sparrows*
Count Morozzo placed successively several full grown
sparrows under a glass receiver, inverted over water. It
was filled with atmospheric air, and afterwards with vital
air. He found,
First —That in atmospheric air, hours. mik#
The first sparrow lived 3
The second sparrow lived .... 3
The third sparrow lived 1
The water rose in the vessels eight lines during the
life of the first; four during the life of the second; and
the third produced no absorption.
Second — In vital air, or oxygen hours, mik.
The first sparrow lived 5 23
The second 3 10
The third 1 30
The fourth ... * 1 .10
IN PNEUMATICS. 933
The fifth 30
The sixth 47
The seventh 27
The eighth 30
The ninth 22
The tenth 21
The above experiments elicit the following conclu-
sions: — 1. That an animal will live longer in vital than
in atmospheric air. 2. That one animal can live in air
in which another has died. 3. That, independent of
air, some respect must be had to the constitution of
the animal ; for the sixth lived 47 minutes, the fifth
only thirty. 4. That there is either an absorption of
air, or the production of a new kind of air, which is
absorbed by the water as it rises.
324 AMUSEMENTS
OPTICS.
DEFINITIONS*
1. WHATEVER grants a passage to light b called
a medium.
2. By rays of light are understood its least parts,
either successive in the same lines, or cotemporary io
several lines.
It is clear that light consists of parts both successive
and cotemporary, because in the same place you may
stop that which comes one moment, and let pass that
which comes immediately after: the least sensible part
which may be stopped, or suffered to proceed, is called
a ray of light.
3. Refran^ibility is that disposition of a ray of light
to be refracted, or turned out of its course, when it
passes out of one medium into another.
When a ray of light passes out of a rarer medium
into a condenser, Sir Isaac Newton supposes that it it
refracted by the superior attraction of the denser me-
dium, and by that means drawn out of its course.
4. Reflexibility is that disposition of a ray of light
to be reflected or turned back into the same medium
from any other medium upon whose surface it may
fall.
Sir Isaac Newton supposes that light is reflected by
impinging upon the solid parts of the body, but by some
power of the body which is evenly diffused all over
its surface, and by which it acts upon the ray, and im-
pels it back without immediate contact.
5. Inflection is that disposition of a ray of light to
be turned out of its course when it passes very near to
the edges of bodies.
IN OPTICS.
225
6. The an'gle of incidence is the angle which the
line described by the incident ray makes with the per-
pendicular to the reflecting or refracting surface at the
point of incidence.
7. The angle of reflection or refraction is the angle
which the line described by the reflected or refracted
ray makes with the perpendicular to the reflecting or
refracting surface at the point of incidence.
8. Any parcel of rays diverging from a point, con-
sidered as separate from the rest, is called a pencil of
rays.
9. A lens is a medium bounded by two spherical,
or one plain and one spherical surface; and the line
joining the centres, or which passes perpendicularly
through each surface, is called the axis.
There are six lenses, a double convex, a double con-
cave, a plano-convex, a plano-concave, a concave-con-
vex, and a meniscus.
10. The focus of rays is that point from which they
diverge, or to which they converge.
The focus of parallel rays is called the principej
focus.
The sun's light consists of rays of different colours,
and differently refrangible.
For if the sun s rays be admitted iilto a dark room
through a small hole in a window shutter, and be re-
fracted through a prism, the image is not round, but a
long figure with parallel sides and semicircular ends,
the length of which is above Ave times its breadth ;
that end which has suffered the least refraction is red,
and that which has suflered the greatest is violet : the
whole image consists of seven distinct colours, lying
in the following order — red, orange, yellow, green,
blue, indigo, violet. The red is the least refrangible,
and the others more in their order. These are called
primary colours, all other colours being only different
combinations of these. Each colour forms a distinct
image of the sun, which images, in this experiment^
running into each other, make a gradual change of
^lour in the image ; but if a convex lens be placed
l5
326 AMUSEIHEMTS
before the prism, each image will be diminished, and
by that means they will be separated, and each ren-
dered distinct.
If two coloured images be formed with two prisms,
and thrown one upon the other, then if that image be
looked at through a prism, the images will be again se-
parated.
The primary colours cannot be separated into other
colours by any refraction.
For if in the last experiment all the colours but one
be stopped, for instance, the red, and that be again
refracted by a prism, it suffers no alteration in colour.
By suffering the colours to pass in succession, from th«
red, each preserves its colour, but the quantity of re-
fraction keeps increasing. The image of each colour is
perfectly circular, which shews that the lights of each
colour is refracted regularly without any dilation of the
rays; it is therefore incompounded, or homogeneal.
If the breadth of each colour in the spectrum formed
by the prism be measured, it will appear that th^
bread-thof.the red, orange, yellow, green, blue, indigo,
riolet, are as the numbers 45, 27, 48, 60, 60, 40, 80,
respectively.
If the circumference of a circle be divided into 45',
27°, 48°, 60°, 60°, 40°, 80°, and the respective sec^
tors be painted red, orange, yellow, green, blue, in-
digo, violet, and the circle be turned swiftly, it will
appear nearly white ; for the ideas we have from the
impression of light remain for a short time, and thus
the colours excite the same sensation as if they all en-
tered the eye collected together.
If the direct image of the sun through a small hole
be received upon a screen perpendicular to the rays,
and the rays be then intercepted' by a prism, and fall
perpendicularly on the first side, if the distance from
the place of the direct image to the nearest edge of the
red and farthest of the violet be measured, they will
be the tangents of the angles of deviation, and the ra-
dius of which is the distance from the point where the
rays emerge to the place of the direct image.
The angle of incidence on the second side of the
IN OPOTCS. S87
I
prism ecjual the refracting angle or the prism, to which
add the deviations of the two extreme colours, and wc
get the two angles of refraction, the sined of which will
be to the sines of incidence as 77 and 78 to 50 : hence,
if th6 difference between 77 and 78 be divided in the
ratio of the breadth of each colour, it gives for the sines
of refraction, the common sine of incidence, being 50:
that is, the sine of incidence. The sine of refraction of
the red rays :: 50: not less than 77, nor greater than
774^, the boundary of the red ; and the same for the
rest.
Candle-light is of the same nature as the light from
the sun; for rays from a candle may be separated
into all the different colours, and they lie in the same
order as in the light from the sun.
The sun's light consists of rays which differ in flexi-
bility, and those rays which ar^, most refrangible are
most reflexible.
For after forming a coloured image, as before, with
a prism, by turning the prism about its axis, until the
Tays within it which, in going out into the air were
refracted at its base, become so oblique to the base as
to begin to be totally reflected thereby, those rays be-
come first reflected which before, at equal incidences
with the rest had suffered the greatest refraction.
According to Sir Isaac Newton, the colours of natu-
ral bodies arise from hence, that some reflect one $ort
of rays, and others another sort more copiously than th^
rest
For every body looks most splendid in the light of its
own colour, and therefore it reflects that the most co-
|>k>usly: besides, by reflection you cannot change the
eolour of any sort of rays ; and as bodies are seen by
reflection, they must appear of the colour of those rays
which they reflect. This is the opinion of Sir Isaac
^K^wtOH ; but Mr. Delaval accounts for the colours of
natural bodies in a manner different from this. See
the Manchester Memoirs, Vol. II.
Thin transparent substances, as glass, water, air,
&c. exhibit various colours according to their thick»
ness.
l6
229 AMUSEMENTS
« For a very thin- glass bubble* or a bubble of water,
, will appear to have concentric colours: the bubU«
blown with water, first made tenacious by dissolving a
little soap in it, continually grows thinner at the t0|i
by the subsiding of the water, the rings of colours di*
lating slowly, and overspreading the whole bubble. A
convex and concave lens of nearly the same curvature
being pressed closely together, exliibit rings of coloun
about the point where they touch. Between the co-
lours there are dark rings, .and when the glasses am
very much compressed, the central spot is dark. ^
Isaac Newton, to whom we owe all these discoveries,
found the thickness of the air between the glasses whert
the tolouis appeared to be as I, 3, 5, 7, 9, &c. and
the thickness where the dark rings appeared to be as 0,
2, 4, 6, 8, &c. the coloured rings must have ap*
peared from the reflection of the light, and the dark
rings from the transmission of the light: the rays
therefore were transmitted when the thickness of <th«
air was 0, 2, 4, 6, 8, &c. and reflected at the thick*
ness 1, 3, 5, 7, 9, &c, Sir Isaac Newton therefore,
supposes, that every ray of light in its passage through
any refracting surface is put into a certain constitution
or state, which in the progress of the ray returns at
equal intervals, and disposes the ray at every return to
be easily transmitted through the next refracting sur-
face, and between the returns to be easily reflected bj
it. These he calls fits of easy transmission and refle<y-
tion.
Three Objects discernible only with both E^es,
If you fix three pieces of paper against the wall ofk
room at equal distances, at the height of your eye,
placing yourself directly before them, at a few yards
distance, and close your right eye, and look at them
with your left, you will see only two of them, suppose
the first and second; alter the position of your eye,
and you will see the first and third ; alter your position
ji second time, you will see the second and thirid, but
never the whole three together: by which it a|>peani.
IN OPTICS. 239
that a person who has only one eye can never see thre«
objects placed in this position, nor all the parts of on*,
object of the same extent, without altering the situa**
tion of his eye.
To construct the Camera Obscura.
Make a circular hole in the shutter of a window from
whence there is a prospect of some distance; in this
hole place a magnifying glass, either double or single,
whose focus is at the distance of five or six feet ; no
light must enter the room but through this glass. At
a distance from it, equal to its focus, place a very white
pasteboard (what is called a Bristol board, if you can
procure one large enough, will answer extremely well ;)
this board must be two feet and a half long, and
eighteen or twenty inches high, with a black border
round it: bend the length of it inward to the form of
part of a circle^ whose diameter is equal to double thfi
focal distance of the glass. Fix it on a frame of the
tame figure, and put it on a moveable foot, that it may
be easily placed at that distance from the glass^ where
the objects appear to the greatest perfection.. When
it is thus placed, all the objects in front of the window
will be painted on the paper in an inverted position^
with the greatest regularity, and in the most natural
colours. If you place a swing looking-glass outside
the window, by turning it more or less, you will have
on the paper all the objects on each side the window.
If, instead of placing the looking-glass outside the
window, you place it in the room above the hole (which
must then be made near the top of the shutter), you
may have the representation on a paper placed hori-
con tally on a table, and draw at your leisure all th#
objects reflected.
.Observe, the best situation is directly north; and
be best time of the day is noon.
930 amosehenM
Tie Magnifying Reflectorl
Let the rays of light that pass through the magnify-
ing glass in the shutter be thrown on a large concave
mirror, properly fixed in a frame. Then take a thin
/itrip of glass, and stick any small object on it; hold it
in the intervening rays at a little more than the focftl
distance from the mirror, and you will see on the oppo-
site wall, amidst the rejected rays, the image of that
object, very large, and beautifully clear and bright
Optical Augmentation,
Take a large drinking glass of a conical figure, that
is, small at bottom and wide at top ; in which pat a
'Shilling, and fill the glass about half full of water ; then
place a plate on the top of it, and turn it quickly oyer,
Uiat the water may not get out. You will then see
on the plate a piece of the size of a half-crown, and
somewhat higher up', another piece of the size of a
shilling.
This phenomenon arises from seeing the piece through
the conical surface of the water at the side of the glass,
and through the flat surface at the top of the water at
the same time; for the conical surface dilates the rays,
and makes the piece appear larger; but by the flat
surface the rays are only refracted, by which the piece
is seen higher up in the glass, but still of its natural
size. That this is the cause will be further evident by
filling the glass with water, for as the shilling cannot
be then seen from the top, the large piece only will be
Tisible.
After you have amused yourself with this remarka-
ble phenomenon, you may gif e the glass to a senrant,
telling him to throw out the water, and take care of
the two pieces of money ; and if he has no suspicion of
the deception, he will be not a little surprised to find
one piece only.
S3I
To magnify tmall objectt by meant of the twi't ra]f* let tnl*
a 4ark Chamber,
Let the ra;s of light th^t pass through the leufl in
the shutter be thrown on a large coacare mirror, pro-
{wrly hxed in a frame; then take a slip, or thin plate
</glasB, and sticking any small object on it, hold it in
the incident rays, at a little more than the focal dii-
tftnce from the mirror, and you will see on the oj^m*
lite wall, amidst the reflected rays, the image of that
' object, very large, and extremely cle^ and bright. —
This experiment nefer fails to give the spectator the
]uj;heit latisfactiou.
The Magic Lantern.
This »ery remarkable machine, which is now known
over all the world, caused great astonishment at iU
origin: It is still beheld with pleasing admiration, and
the spectator very frequently contents himself with
.232 AMUSEMENTS.
wondering at its effects, without endeavouring to inves-
tig^te their cause. The invention of this ingenious il-*
lusion is attributed to the celebrated P. Kircher, who
has published, on various sciences, works^ equally
learned, curious, and entertaining.
The design of this machine is to represent at large,
<m a cloth or board, placed in the dark, the images of
small objects, painted with transparent colours on
plates of glass.
Its construction is as follows: Let AB C D (see th«
figure) be a tin box, eight inches high, ten long, and six
wide (or any other similar dimensions); at the top must
be a funnel, E, of four inches diameter, with a cover,
which at the same time that it gives a passage to the
smoke, prevents the light from coming out of thm
box.
On the side A C, there is a door, by which is adjusted
a concave mirror, G, of metal or tin, and of five incites
diameter ; being part of a sphere whose diameter is 18
inches, tliis mirror must be so disposed that it may l>e
pushed forward or drawn back by means of the handle
H, that enters the tin tube I, which is soldered to tlM
door.
In the middle of the box must be placed a low tin
lamp, K, which is to be moveable. It should havs
three or four lights, that must be at the height of Um
focus of the mirror G.'
In the side B D, and opposite to the mirror, then
muKt be an aperture of three inches wide, and two
inches and a half high, in which is to be fixed a conrez
glass L, of the same dimension. '' I prefer this form
for the glass,'' says M. Guyot, that the picture thrown
upon the cloth may have the same form, which is muck
preferable to a circular aperture, through which the
figures can never be completely seen but when they are
at the center of the glass." It is surprising that this
imperfection has been suffered to continue so long,
when it is so easily remedied. The focus must be from
four inches and a half to five inches, so that the lamp
may be placed both in its focus, and in that of the eoo-
cave mirror.
IN OPTICS. 233
On the same side place a piece of tin MN, of four
inches and a half square, having an opening at the
sides about four inches and a half high, and a quarter
of an inch wide; through this opening or groove are to
pass the glasses, on which are painted the figures that
are to be seen on the cloth. In this tin piece, and op-
posite the glass L, let there be an aperture of three
inches and a quarter long, and two inches and a quar-
ter high, to which must be adjusted a tube O, of the
same form, and six inches long. This tube is to be fixed
into the piece M N. Another tube, six inches long,
and moveable, must enter that just mentioned, in which
must be placed two convex lenses, P and Q ; that of P
may have a focus of about three inches, and that of Q,
which is to be placed at the extremity of the tube, one
of ten or twelve inches. The distance between these
glasses is to be regulated by their foci. Between these
glasses there must be placed a pasteboard R, in which
18 an aperture of an inch wide, and 4-5ths of an inch
high : — by placing this tube farther in or out of the
other, the images on the cloth will appear larger or
smaller.
From what has been said ofthe preceding machines,
the construction of this will be easily understood. The
foci of the concave mirror, and the lens L, meeting in
the flame of the lamp, they together throw a strong
light on the figures painted on the glasses that pass
through the groove M N, and by that means render
their colours distinct on the cloth. The rays from those
glasses passing through the lens P, are collected by the
aperture in the pasteboard R, and conveyed to the lens
Q, by which they are thrown on the cloth.
The lantern being thus adjusted, you must provide
plates of clear glass, of twelve or fifteen inches long,
and three inches wide, which are to be placed in thi-i
frames, that they may pass freely through the groove
M N, after being painted in the manner we shall now
describe.
234 AMUSEMENTS
Method of painting the glasses for the Lantern,
Draw on a paper the subject you intend to paint, and
fix it at each end to the glass ; provide a varnish with
which you have mixed some black paint, and with a
fine pencil draw on the other side of the glass, with
very light touches, the design drawn on the paper. If
you are desirous of making the painting as perfect as
possible, you should draw some of the outlines in their
proper colours, provided they are the strongest tints
of those colours that are used. When the outlines are
dry, you colour the figures with their proper tints or
degradations; and those cqlours will not peel off, if
you temper them with a strong white varnish. All those
colours that are not terestrial, as Prussian blue, car-
mine, calcined verdigris, &c. may be used to advan-
tage, when tempered with a proper varnish. You are
then to shade them with black mixed with the same
Tarnish, or white bistre, as you find convenient; you
may also leave strong lights in some parts without any
colours, in order to produce a more striking effect. —
Observe, in particular, not to use more than four or five
colours, such as blue, red, green, and yellow. You
should employ ,• however, a great variety of tints, to
give your painting a more natural air, without which
they will represent vulgar objects, which are by no
means the more pleasing because they are gaudy.
When the lamp in this lantern is lighted, and bj
drawing out the tube to a proper length, the figures
painted on the glass appear bright and well defined^
tlie spectator cannot fail of being highly entertained
by the succession of natural or grotesque figures that
are painted on the glasses.
This piece of optics may be rendered much more
amusing, and, at the same time, more marvellous, bj
preparing figures to which different natural motions
may be given. There are, in the Philosophical Essays
of M. Muschenbrock, different methods of performing
all these various movements^ by some mechanical
IN OPTICS. 235
contrivances that are not difficult to execute, whick
every one may perform according to his own taste; —
either by movements in the figures themselves, or by
painting the subject on two glasses, and passing them
at the same time through the groove.
236
AMUSING SECRETS.
To make a ring be suspended by a thread after it has been
burnt,
NOTHING is necessary for this purpose, but to em-
ploy a thread which has been soaked in a solution of
common salt in river water. Though flame be applied
to the thread, it will still have strength sufficient to
sustain the ring.
To make people in a room hate a hideous appearance^
Dissolve salt in an iiifusion of saffron in spirit of
wine; then dip some tow in the solution, and having
set fire to it, extinguish the other lights in the room.
To form figures in relief on an egg.
Delineate on the shell any figures at pleasure, with
melted tallow, or any other fat oily substance, proof
against acids ; then immerse the egg in strong vinegar,
and let it remain till the acid has sufHciently corroded
that part of the shell not covered with the tallow or
oil.
To change a colour from white to blue.
Dissolve copper filings in a phial of volatile alkali :
when the phial is unstopped, the liquor will be bluish;
but when unstopped it will be white.
AMUSING SECRETS. 237
To make a red Hquory wkichy when poured into different
glasses, shall become yellow, blue, Mucky or purple.
This phenomenon may be produced by the following
process: Infuse a few shavings of logwood in com-
mon water, and when the liquor is sufficiently red,
pour it into a bottle. Then take three drinking glasses
and rinse one of them with strong vinegar; throw
into the second a small quantity of pounded aluni,
which will not be observed if the glass has been newly
washed, and leave the third without any preparation.
If the red liquor in the bottle be poured into the first
glass, it will assume a straw colour, somewhat similar
to that of Madeira wine; if into the second, it will pass
gradually from bluish gray to black, provided it be
stirred with a bit of iron, such as a key, for example,
which has been privately immersed in good vinegar. —
In the third glass, the red colour will assume a violet
tint.
To make pomatum with water and wax, two substances
which do not combine together.
Put into a new glazed earthen pot six ounces of river
water, and two ounces of wax, in which, to render the
process more marvellous, you must have concealed a
strong dose of salt of tartar. If the whole be then ex-
posed to a considerable degree of heat, it will assume
the consistence of pomatum, and may be used for
cleansing the skin.
How a body of a combustible nature may be penetrated by
Jire without being consumed.
Put into an iron box a piece of charcoal, sufficient
to fill it entirely, and solder on the lid. If the box be
then thrown into the fire, it will become red, and it
may even be left in it for several hours or days. When
opened, after it has cooled, the charcoal will be found
entire, though there can be no doubt of its having been
penetrated by the matter of the fire, as well as the
whole metal of the box which contains it.
%36 AMmWB MSeRCM".
Apparent transmutation of iron into copper or silver.
Dissolve blue vitriol in water, till the latter is nearly
saturated, and immerse into the solution small plates
of iron or coarse filings of that metal. These small
plates of iron or filings will be attacked and dissolved
by the acid of the vitriol, while its copper will be pre-
cipitated and deposited in the place of the iron dis-
solved, it will be so completely covered with cupreous
particles, that it will seem to be converted into cop-
per. This is an experiment commonly shewn to those
who visit copper mines. In Savoy keys have been seen
to become entirely of a copper colour, after being im-
mersed some minutes in water, collected at the bottom
of a copper-mine.
If you dissolve mercury in marine acid, and immerse
in it a bit of iron ; or if the solution be rubbed over
the iron, it will assume a silver colour. Jugglers some-
times exhibit this chemical deception at the expence
of the credulous and ignorant.
hemark.
In this case there is no real trangmntatiou, but only
the appearance of one. The iron is not changed into
copper ; the latter, held in solution by the liquor im-
pregnated by the vitriolic acid, is only deposited in the
place of the iron with which the acid becomes charged,
while it abandons the copper. Every time, indeed,
that a menstruum, holding any substance in solution,
is presented to another substance which it can dissolve
with more facility, it abandons the former, and becomes
charged with the second. This is so certain, that
when the liquor which has deposited the copper is
evaporated, it produces crystals of green vitriol, which,
as is well known, are formed by the combination of the
acid with iron. This process is indeed practised, on a
large scale, in the mines of Savoy. The liquor in
question, which is nothing but a pretty strong solution
of blue vitriol, is put into casks, or large square reser-
voirs; pieces of old iron being then immersed in it, are
£10 AMUSING SECRETS.
•ited a greater or less quantity of calcareous eartl
The manner in which this is done, is as follows :
Water, in general, is hard only because it holds ii
solution selenite or gypsum (a combination of vitrioli
acid with calcareous earth), which it has dissolved i
its passage through the bowels of the earth, or which
has been formed by the water first becoming impreg^
nated with vitriolic salts, and afterwards in its course
meeting with, and dissolving, a portion of calcareoui
earth.
On the other hand, soap is an artificial combina-
tion of mixed alkali with oil, or with some other greasy
substance, and which have no great affinity.
When soap, therefore, is dissolved in water impreg-
nated with selenite, the vitriolic acid of the latter hav-
ing. a greater tendency to unite with the fixed alkali,
than with the calcareous earth, which enter intp the
composition of the selenite, abandons that earth, and
combines with the fixed alkali in such a manner, that
the soap is decomposed; and, as the oil is immiscible -
with water, it is diffused through it in the form of white
flakes, while the calcareous earth of the selenite falls
to the bottom.
Bj/ the mixture of two transparent liquors, to produce a
blackish liquor : — Method of making good ink.
Provide a solution of green or ferruginous vitriol,
and an infusion of gall-nuts, or any other astringent
vegetable substance, such as oak-leaves, well clarified
and filtered; if you then pour the one liquor into the
other, the compound will immediately become obscure,
and at last black.
If the liquor be suffered to remain at rest, the black
matter suspended in it will fall to the bottom, and leave
it transparent.
REMARK.
This experiment may serve to explain the formation
of common ink ; for the ink we use is nothing but a
AMUSING SECRETS. 241
solution of green vitriol mixed with an infusion of gail-
auts, and a little gum. The blackness arises from the
property which the gall-nuts have of precipitating, of
it black or blue colour, the iron held in solution by the
water impregnated with vitriolic acid ; but as the iron
would soon fall to the bottom, it is retained by the
addition of gum, which gives to the water sufficient
viscosity to prevent tfie iron from being precipitated.
The reader, perhaps, will not be displeased to find
here the following recipe for making good ink: —
Take one pound of gall-nuts, six ounces of gum
arabic, six ounces of green copperas, and one gallon
of common water or beer; pound the gall-nuts, and
infuse them in a gentle heat for twenty-four hours,
without bringing the mixture to elbulUtion ; then add
the gum in powder^ When the gum is dissolved, put
in the green vitriol. And if you then strain the mix-
ture, you will obtain very fine ink.
To produce inflammable and fulminating vapours.
Put into a moderately sized bottle, with a short wide
neck, three ounces of oilor spirit af vitriol, with twelve
ounces of common water, and throw into it at different
times, an ounce or two of iron filings. A violent effer-
vescence will then take place, and white vapours will
arise from the mixture. If a taper be presented to the
mouth of the bottle, these vapours will inflame and
produce a violent detonation, which may be repeated
several times, as long as the liquor continues to furnish
similar vapours.
The Philosophical Candle.
Provide a bladder, into the orifice of which is inserted
a metal tube, some inches in length, that can be
adapted to the neck of a bottle, containing the same
mixture as that used in the preceding experiment.
Having then. suffered the atmospheric air to be ex-
pelled from the bottle, by the elastic vapour produced
by the solution, apply to the mouth of it the orifice o#
S43 AlfUSING SECRETS.
the bladder, after carefidly expressing from it the com-
mon air (which you must not fail to do^ or the bladd^
will explode). The bladder by these means will be-
come mled with the inflammable air; which, if yoa
force out against the flame of a taper, by pressing the
sides of the bladder, will forn^ a jet of beautiful green
flame. This is what the chemists call a philosophical
candle.
To make an Artificial Volcano.
For this curious experiment, which enables us to
assign a very probable cause for volcam^es, we are in-
debted to Lemery.
Mix equal parts of pounded sulphur and iron filings,
and having formed the whole into a paste with water,
biiry a certain quantity of it, forty or fifty pounds, for
example, at about the depth of a foot below the sur-
face of the earth. In ten or twelte hours after, if the
weather be warm, the earth will swell up and burst,
and flames will issue out, which will enlarge the aper-
ture, scattering around a yellow and blackish dust.
It is not impossible that what is here seen in mi-
niature, takes place on a grand scale in yolcanoes ; as
it is well known that they always furnish abundance
of sulphur, and that' the matters they throw up abound
in metallic and probably ferruginous particles; for
iron is the only metal which has the property of pro-
ducing an efferve&ence with sulphur, when they are
mixed together.
But it may easily be conceived, from the efiect of a
small quantity of the above mixture, what thousands
or millions of pounds of i^ would produce : there is
no doubt that the result would be phenomena as ter-
rible as those of earthquakes, and of those volcanic
eruptions with which they are generally accompanied.
To make Fulminating Powder,
Mix together three parts of nitre, two of well-dried
fiiced alkali, and one of sulphur: if a little of this mix-
AMUSING SECRETS. 343
ture be put into an iron spoon, over a gentle fire, ca-»
pable however of melting the sulphur; when it acquires
a certain degree of heat, it will detonate with a loud
noise, like the report of a small cannon. .
This would not be the case if the mixture were ex-
posed to a heat too violent: the parts only most ex-
posed to the fire would detonate, and by these means
the effect would be greatly lessened.
If thrown on the fire, it would not detonate, and
would produce no other effect than pure nitre, which
indeed detonates, but without any explosion.
To farm a combination which when cold is liquid and
transparent y but when warm beomnes thick and opake,
r
Put equal quantities of fixed alkali, either mineral or
vegetable, and of well-pulverised quick lime, into a
sufficient quantity of water, and expose it to strong
and speedy ebullition. Then filtre the product, which
at first will pass through with difficulty, but afterwards
with more ease, and preserve it in a bottle well
stopped. This liquor, when made to boil, either in the
bottle or in any other vessel, will become turbid^ and
cussume the consistence of very thick glue ; but when
cold, it will recover its fluiditly and transparency.
To make a flash, like that of lightnings appear in a room
when any one enters it with a lighted candle.
Dissolve camphor in spirit of wine, and depodt the
vessel containing the solution in a very close room,
where the spirit of wine must be made to evaporate by
speedy and strong ebullition. If any one then enters
tlie.room with a lighted candle, the air will inflame,
while the combustion will be so sudden, and of so
short a duration, as to occasion no danger.
It is not improbable that the same effect might be
produced, by filling the air of an apartment with the
dust of the seed of a certain kind of lyooperdon, which
it inflammable.
m2
244 AMUSING SECRETS.
()f Sympathetic Inks y and some tricks which may be per"
formed by means of them,
■ Sympathetic inks are certain liquors, which alone,
and in their natural state, are colourless ; but which,
by being mixed with each other, or by some particular
circumstance, assume a certain colour.
Chemistry presents us with a great many liquors of
this kind, the most curious of which we shall here de-
scribe.
1st. If you write with a solution of green vitriol, to
which a little acid has been added, the writing will be
perfectly colourless and invisible. To render it visible,
nothing will be necessary but to immerse the paper in
an infusion of gall-nuts in water, or to draw a sponge
moistened with the infusion over it.
2rd. If you are desirous of having an ink that shall
become blue, you must write with an acid solution of
green vitriol, and moisten the writing with a liquor pre-
pared in the following manner :
Make four ounces of tartar, mixed with the same
quantity of nitre, to detonate on charcoal ; then put
this alkali into a crucible with four ounces of dried ox
blood, and cover the crucible with a lid, having in it
only one small aperture. Calcine the mixture over a
moderate fire, till no more smoke issues from it, and
then bring the whole to a moderate red heat ; take the
matter from the crucible, and immerse it, while still
red, in two quarts of water, where it will dissolve by
ebullition ; and when the liquor is reduced to one half,
it will be ready for use. If you then moisten with it
the writing above mentioned, it will immediately as-
sume a beautiful blue colour. In this operation, instead
of black ink, there is formed Prussian blue.
3rd. If you dissolve bismuth in nitrous acid, and
write with the solution, the letters will be invisible. To
make them appear, you must employ the following
liquor :
Boil a strong solution of fixed alkali with sulphur
reduced to a very fine powder, until it dissolves as
much of it as it can ; the result will be a liquor which
AMUSING SECRETS. 245
exhales vapours of a very disagreeable odour, and to
which, if the above writing be exposed, it will become
black.
4th. Of all tbe different kinds of sympathetic ink^
the most curious is that made from cobalt. It is a very
singular phenomenon, that the characters or figures
traced out with this ink, may be made to disappear
and re-appear at pleasure. This property is peculiar
to ink made with cobalt; for all the other kinds are at
first invisible, until some substance has been applied
to make them appear : when they have once appeared^
they remain.
To prepare this ink, take zafier, and dissolve it in
aqua regia (nitro milriatic acid) till the acid extracts
from it every thing it can ; that is to say, the metallic
part of the cobalt, which communicates to the zafier a
blue colour; then dilute the solution, which is very
acrid, with common water. If you write with this li-
quor on paper, the characters will be invisible; but
when exposed to a sufficient degree of heat they will
become green. When the paper has cooled, they will
disappear.
It must, however, be observed, that if the paper be
heated too mu€h, they will not disappear at alL
REMARK.
With this kind of ink some very ingenious and
amusing tricks, such as the following, may be per-
formed.
1st. To make a Drawing, which shall aUemately represent
Winter and Summer.
Draw a landscape, and delineate the ground, and the
trunks and branches of the trees, with the usual colours
employed for that purpose, but the grass and leaves of
the trees with the hquor above mentioned. By these
means you will have a drawing, which, at the common
temperature of the atmosphere, will represent a wint«-
piece ; but if it be exposed to a proper degree of heat|
M 3
246 AMUSING SECEETS.
not too strong, you will see the ground become co-
, yered with verdure, and the trees with leayes, so as to
present a view in summer.
Screens painted in this manner, were f<M*merly made
at Paris. Those to whom they were {Nresented, if mi-
acquainted with the artifice, were astonished to find,
when they made use of them, that the yiew9 they exhi-
bited were totally changed.
2nd. The Magic Oracle.
Write on several sheets of paper, with common ink,
a certain number of questions, and below each ques-
tion write the answer with the above kind of sympa-
thetic ink. The same questions must be ¥n:itten on
several pieces of paper, but with different answers, that
the artifice may be better concealed.
Then provide a box, to which you may give the name
of the Sybil's cave, or any other at pleasure, and con-
taining in the lid a plate of iron made very hot, in order
that the inside of it may be heated to a certain de-
gree.
Having selected some of the questions, take the bits
of paper containing them, and tell the company that
you are going to send them to the Sybil, or Oracle, to
obtain an answer ; introduce them into the heated box,
and when they have remained in it some minutes, take
them out, and shew the answers which have been
written.
You must, however, soon lay aside the bits of paper;
for if they remain long in the hands of those to whom
• the trick is exhibited, they would see the answers gra-
dually disappear, as the paper becomes cold.
Of Metallic Vegetations.
To see a kind of shrub rise up in a bottle, and even
throw out branches, and sometimes a kind of firuit, is
one of the most curious spectacles exhibited by che-
mistry. The operation by which this delusive image is
produced, has been called chemical or metalic vegeta*
AMUSING SECRETS, 247
tioD, because performed by means of metallic sub-
stances ; and it is not improbable, that some respecta-
ble persons, who thought they saw a real palingenesy>
have been deceived by a similar artifice. However this
may be, the following are fhe most curious of these ve-
getations, which in fact are only a port of crystal-
lizations.
Arbor Martisy or Tree of Mars^
Dissolve iron filings in spirit of nitre (aqua fortis))
moderately concentrated, till the acid is saturated ; then
pour gradually into the solution a solution of fixed al-
kali, commonly called oil of tartar per deliquium. A
strong efiervescence will take plade, and the iron, in-
stead of falling to the bottom of the vessel, will after-
wards rise, so as to cover its sides, forming a multitude
of ramifications heaped one upon the other, which will
sometimes pass over the edge of the vessel, and extend
themselves on the outside, with aH the appearance of a
plant. If any of the liquor is spilt, it must be care-
fully collected, and be again put into the vessel, where
it will form new ramifications, which will contribute to
increase the mass of the vegetation.
Arbor DiaruB, or Tree of Diana,
This kind of vegetation is called the Tree of Diana,
because it is formed by means of silver, as the former
is called the Tree of Mars, because produced by iron.
Mix together two parts of very pure mercury, and
four of fine silver, in filings or scsues, by m^ans of tri-
turation with an ivory pesde in a porphyry mortar; then
dissolve this amalgam in four ounces of very pure spi-
rit of nitre, moderately strong, and dilute the solution
with about a pound and a half of distilled water ; shake
the mixture, and preserve it in a bottle well stepped.
Pour an ounce of this liquor into a glass, and throw
into it a smaU bit, about the size of a pea, of an amal-
gam of mercury and silver, similar to the former, and
of the consistence of butter. Soon after you will see
m4
d48 AMUSING SECRET9.
rising from the ball of amalgam, a multitude of small
filaments, which will visibly increase in size, and,
throwing out branches,^ will foitn a sort of shrubs.
The Lead Tree,
This is a more modem invention, and may be pro-
duced by the following very easy method :
To a piece of zinc fasten a wire, crooked in the form
of the worm of a still; let the other end of the wire be
thrust through a cprk. You then pour spring water
into a phial or decanter, to* which you add a small
quantity of sugar of lead; thrust tiie zinc into the bot-
Ue, and with the cork at the end of the wire fasten it
up. In a few days the tree will begin to grow^ and
produce a most beautiful effect.
Nori'metaUic Vegetation.
Cause to decripitate, on burning charcoal, eight
ounces of saltpetre, and place it in a cellar, in order
that it may produce oil of tartar per deliquium, then
gradually pour over it^ to complete saturation, good
spirit of vitriol, and evaporate all the moisture* The
result will be a white, compact, and very acrid saline
matter. Put this matter into an earthen dish, and having
poured over it a gallon of cold water, leave it exposed
to the open air. At the end of some days the water
will evaporate, and there will be formed all around the
vessel ramifications in the form of needles, variously in*
terwoven with each other, and about 15 line» in length.
When the water is entirely evaporated, if more be
added, the vegetation will continue.
It may be readily seen^ that this is nothmg but the
mere crystallization of a neutral salt, formed by the
vitriolic acid and the alkali of the nitre employed, that
is to say, vitriolated tartar.
AMUSING SECRETS. 249
To produce Heat, and even Tlame, by means of two cold
Licpiors,
Put oil of guaiacum into a bason, and provide some
spirit of nitre, so much concentrated, that a small bot-
tle, capable of holding an ounce of water, may contain
nearly an ounce and a half of this acid, make fast
the bottle containing the acid, to the end of a long
stick; and, after taking this precaution, pour about
two thirds of the acid into the oil in the bason ; the
result will be a strong effervescence, which will be fol-
lowed by a very large flame. If an inflammation does
not take place in the course of a few seconds, you have
nothing to do but to pour the remainder of the nitrous
acid over the blackest part of the oil ; a jQiame will then
certainly be produced, and there will remain, after the
combustion, a very large spongy kind of charcoal.
Oil of turpentine, oil of sassafras, and every other
kind of essential oil, may be made to inflame in the like
manner.
The same phenomenon may be produced with fat
oils, such as olive oil, nut oil, and others extracted by
expression, if an acid, formed by equal parts of the vi-
triolic and nitrous acids, well concentrated, be poured
into them.
To fuse Iron in a moment, and make it run into Drops.
Bring a bar of iron to a white heat, and then apply
to it a roll of sulphur ; the iron will be immediately
fused and run down in drops. It will be most conve-
nient to perform this experiment over a bason of water,
in which the drops that fall down will be quenched.—
On examination they will be found reduced into a kind
of cast iron.
This process is employed for making shot used in
hunting ; as the drops, by falling in the water, natu-
rally assume a round form.
m5
250 AMUSING UEfKElS.
Cement for mending broken China,
' Calcine ojster-shellsj and having pounded them, sift
them through a silk sieve, and grind them on por-
phyry^ till Uiey are reduced to an impalpable powder.
Then take the whites of several eggs, according td the
quantity of the powder to be used, and form them with
the powder into a kind of paste or glue. With this
paste join the fragments of the porcelain, and press
them together for the space of seven or eight minutes.
No longer time is necessary to dry this mastic ; which
will stand both heat and water, and which will never
g^ve way, even if &e article by any accident should
have a fall.
Process for whitening Prints,
Paste a piece of papev to a very smooth table, that
the boiling water used in the operation may not acquire
a colour, which might lessen ite success. When this
precaution has been taken, spread out the print on the
table, and sprinkle it with boiling water, taking care
to moisten it thoroughly throughout, by means of a fine
sponge. After this process with boiling water has
been repeated three or four times, you will observe the
stains or spots extend themselves ; but this need excite
no uneasiness, as it is only a proof that the dirt im-
bibed by the paper begin& to be dissolved.
After this preparation, the prints must be put into a
copper or wooden vessel, of such a size as to admit of
their being freely stretched out in it; they are then to
be covered with a boiling lye of pot-ash, and care must
be taken to keep it hot as long as possible. After the
whole has cooled, take out the prints with care ; spread
them on stretched cords, and when half drv, press them
between leaves of paper, in order that they may not
contract wrinkles.
By this process, spots and stains of every kind may
be removea. *
AMUSING SECRETS. 251
Method of taking Paintings from the old Canvass^ and
transferring them to New,
Take the painting from its frame, and tack it down
on a very smooth table, with the face upwards, and in
such a manner that it may be well stretched, and free
from wrinkles ; then cover it with a stratum of strong
glue, and lay over it some sheets of large white paper,
of the strongest kind you can procure. When the
whole has dried, draw the tacks, and having inverted
the painting, that is, turned the back uppermost, with*
out fixing it, dip a sponge in tepid water, and gra-
dually moisten the canvass, trying it from time to time
at the edges, to see whether it begins to detach itself
from the painting.
When you find it sufficiently loose, detach it care-
fully along one of the edges, and fold back the part so
detached : if you then roll it with both hands, the
whole canvass may, by these means, be removed. —
When this is done, wash well the back of the j^ainting
with a sponge dipped ih water, until all the old size has
been nearly removed ; then cover the back of the paint-
ing with a new stratum of size, or the usual priming
applied to new canvass, intended for pictures, and im-
mediately spread over it a new piece t)f canvass, which
jnust be somewhat larger than the painting, in order
that it may be properly stretched and nailed down at
the edges. In the last place, do over the canii^s,
portion by portion, with a stratum of glue, taking care
to spread it with a painter*s mullet, so that it may pass
through the pores of the cloth to the painting.
When the painting is dry, remove it from the table,
and put it into its ilrame, after which you must thorougly
moidt^n the pa^ with, a sponge dipped in warm
water, that it litay be taken ioff withcHil leaving any
traeeft behkid rit, uid . ta wiash out ahy. stains that may
still remain bn ^e painting* CJonoKide the process by
rubbing over the painting with pure nut oil, and when
dry, witli tbe white of an ^gg properly beat up,
h6
253 AMUSING SECRElf.
To^U a Glass with xoater in such a manner Jhat a penan
shall not be able to remove it without spilling it all.
Lay a bet with any one that you will fill a glass
with water, and place it on a table in such a manner
that it cannot be removed without spilling the wh(rfe
water it contains. Then fill a glass with water, and
placing over it a bit of paper, so as to cover the water
andi the edge of the glass; clap the palm of your hand
on the paper, and laying hold of the glass with the
other, suddenly invert it on a very smooth table. If
you then genUy draw out the paper, the water will
remain suspended in the glass, and it will not be pos-
sible to remove it, without spilling the water entirely.
To construct two Figures, one of which shall bkm out a
Candle^ and the other light it again.
Prepare two figures, of any materials whatever, aqd
insert into the mouth of each a tube of the size of a
small quill. Pat into one of these tubes a small piece
of phosphorus, and into the other a few grains of gun-
powder, taking care that each may be retained in the
tube by a bit of paper. If the second figure be applied
to the fiame of a taper, it will extinguish it, and the
first applied will light, it again.
The same kind of phosphorus may be employed, on
the point of a knife, to light a candle which has bee
newly extinguished.
Japan Vases.
The Japanese have the art of making a kind of vases
with the shaving of paper, or with saw-dust, which
when covered with varnish, are capable of containing
hot or cold liquors. These vases, which are' exceed-
ingly neat and light, are ornamented in an agreeable
manner with flowers, birds, and animals, and with
gilt borders.
This preparation is called /Mi^'er maehi^ and is made
of the shavings of white or brown paper, boiled in
AMUSING 8£CRI?rs. 253
water^ and beat in a mortar till they are reduced to a
kind of paste. This paste is afterwards boiled with a
solution of gum arable, to give it tenacity, and by being
pressed into moulds, nibbed over with oil, it may be
formed into toys of various kinds; which when dry
are done over with a mixture of glue and lamp-black,
and then varnished.
The black varnish used for these toys is prepared in
the following manner :
Dissolve, in a glazed earthen pot, a little colopho-
nium, or boiled turpentine, till it becomes black and
friable, and gradi^ally throw into the mixture three
times as much amber finely pulverized; adding from
time to time a little spirit or oil of turpentine. When
the amber is dissolved, besprinkle the mixture withthe
same quantity of sarcocolla gum, continually stirring
the whole, and add spirit of wine till the composition
becomes fluid ; then strain it through a piece of hair-
cloth, pressing it between two boards. This varnish,
when mixed with ivory black, is applied in a warm plac^
on the dried paste of the paper shavings; the articles
are then put into a hot stove, next day removed into a
hotter stove, and the third into one still hotter : each
time they are left till the stove has cooled. The paste,
when, thus varnished, is hard, brilliant, and durable,
and capable of containing liquors either hot or cold.
To construct a Vessel from which water shall escape
through the bottom^ as soon as its mouth is unstopped.
Among the number of amusing tricks, founded oh
philosophical principles, we may class the following.
Provide a vessel of tin-plate, two or three inches in
diameter, and five or six inches in height, having a
mouth about three lines in width, and in the bottom
several small holes, of such a size as to admit a small
needle. Immerse this vessel in water, with its month
.0]^n,'and when fall stop it very closely. If you are
desirous of playing a tric^ to any person, give him this
Te^el) and desire him to unstep it; if ]be does so.
^54 AMUSING 8ECEETS.
placing it on his knees, the water will escape thnmgfa
the holes in the bottom^ so that he will soon be all over
wet
Transparencies.
Those transparencies exhibited on the stage, and
during public festivals, which are illuminated by a light
placed behind them, are prepared in the following
manner. A piece of strong linen or silk, stretched on
a wooden frame, is done oyer with a solution of wax
in oil of turpentine, and during the operation a chaffing
dish is placed below it, that the liquid may be every
•where equally diffused. Any figures at pleasure are
then delineated on the cloth with oil colours, mixed up
with spirit'of turpentine.
Moveable transparencies, exceedingly amusing, may
•be formed in the following manner:
Affix the transparency to a very light circular frame,
supported by an axis on which it can freely turn. The
upper end of the cylinder must be closed by a circular '
piece of tin-plate, cut into inclined planes, like the
ventilators constructed in windows to prevent smoke :
if a lamp be then placed within the cylinder, it will
illuminate the transparency, and at the same time
make it turn round by the means of the current of air
which falls on the tin-plate.
The figures exhibited by this transparency may be
varied a thousand ways, according to the taste of the
artist. They may be made to represent serpents twist*
ing around a column, &c.
It is by the same mechanism that a spiral piece of
card or paper, placed on a stove, turns round of itself,
and serves as a thermometer to regulate .the heat
Method qf fixing Crayons.
. Crayon painting is superior to oil painting in bright-
ness, freshness, splendour of colounng, and fidelity of
likeness^ It is attended with this advantage mo,
i.MU8ING SECRETS. 255
fiiatitisnot subject to that reflection of light which '
preyents .the beauty of a painting from being seen ex-
cept from a certain point of view. On account of these
valuable qualities, it would certainly have been pre-
ferred to oil painting, had it been equally durable ; but
it has this inconvenience, that it is liable to be de-
stroyed by the least friction. At the end of a few
years master-pieces of this kind perish, because the
powder of the crayons detaches itself, or becomes
mbuldy , especially if great care be not taken to preserve
these paintings from moisture, and froin the heat of
the sun. The following liquor, however, has been em-
ployed with success for fixing crayons : it is not ex-
pensive, and nothing is necessary but to hnmerse the
painting in it for a few moments.
To prepare this liquor, dissolve Ronian alum pul-
verised, in two glassfulls of very pure water, and wtien
the water is saturated, decant it from off the alum,
which may have remained undissolved at the bottom
of the vessel. This observation is of ereat import-
ance ; for if the alum which has not been dissolved were
left in the liquor, by becoming dry it might tarnish the
painting, and produce whitish spots in those places
where the liquor accumulates itself in draining off. —
Into this water, well impregnated with alum, put a
small quantity of very transparent and pure fish glue,
leaving it to dissolve for twenty-four hours, and then
boil the whole, that the glue may be dissolved com-
pletely. The liquor must afterwards be strained through
a piece of linen, to free it from any impurities it may
contain.
In the last place, pour the water, thus impregnated
yrith alum and glue, into a bottle containing three pints
of brandy, not coloured, and mixed with a large glass-
full of spirit of wine. A greater or less quantity of
this liquor may be made according to the size of the
paintings to be fixed, prdvided care be taken to in«
crease the ingredients in the proper proportions. It is,
however, to be observed, that it must not be used
when too old, as in that case it would weaken the
f plendour of Uie paintkig.
256 AMUSING SECRETg*
Put the liquor, thus prepared, into a vessel of lead,
or of any other substance, so large that the painting
may be immersed in it, and heat it in a balneum ma-
risi, taking care that the fish-glue be well dissolved;
for before the liquor is heated, especially if the weather
be cold, it will deposit itself at the bottom. Place in
each comer of the bason a bit of lead, in such a man-
ner, that the liquor may rise over it no more than a line
at most, and then lay hold of the painting, keeping it
in an horizontal position, and immerse it gently into
the liquor. The pieces of lead, placed in the vessel,
will prevent it from sinking too deep. The time em-
ployed in immersing and ta^ng out the painting, ought
not to exceed a second.
The painting must be taken out horizontally^ and be
deposited in &e same position in some place where it
can rest on its two borders, which it will do if sup-
ported by two chairs.
If the above process be properly followed, it will be
found that all the tints have retained their original
freshness and primitive colour. Crayons fixed in this
manner, will bear even to be covered with a varnish,
which may supply the place of glass. To lay on this
yamish, the following method may be employed.
When the painting is fixed and dry, apply over it,
with a soft brush, a stratum or two of melted fish-glue,
mixed with about a third of spirit of wine, and sufiB-
ctently strong, that when cold it may form a sort of
jelly. When this preparation is dry, apply that varnish
used for varnishing pnnts, which will produce the same
effect as on paintings in distemper.
Crayon paintings, fixed in the above manner, are at-
tended with this advantage, that they may be re-
touched; for the crayons will make an impression as
before; some strengthening touches may even be added
with colours in distemper. This method employed for
crayons, may be used also for fixing chalk drawings.
AMUSING SECRETS. 257
A curious Illusion^
Pill a glass goblet with pure water, and put into it
a piece of money, such, for example, as a shilling;
then cover the goblet with a plate, and laying your
hand upon the latter, invert the whole speedily, so that
the air not having time to enter, the water may not be
able to escape.
If you look at the piece of money, which will then be
on the plate, it will appear the size of a half-crown,
and it will be seen also of its real size a little above
the former image, which will make those unacquainted
with the singular effects of refraction to beUeve, that
there are really below the goblet, a half-crown and a
shilhng. When the goblet is removed the illusion will
cease.
An Object being placed behind a Convex Glass, to make it
appear before it.
Provide any object, such, for example, as a small ar-
row of wood, an inch and a half in length, and tie it
perpendiculatly to a piece of black card, which must
be suspended from a wall, at about the height of the
eye. Throw a strong light on the card, and place be-
fore it a lenticular glass, two or three inches in diame-
ter, in such a manner, that it may be distant from the
arrow about twice the length of its focus. If you then
make a person stand at a proper distance, opposite to
the glass, the arrow will appear to him to be suspended
in the air before the glass.
It is evident, that this singular effect of dioptrics,
with taste and a little ingenuity, may be applied to a
variety of other amusements, which it is needless here
to detail.
The Chinese Shadows. Ombres Chinoises.
Make an aperture in a partition wall, of any size,
for example, four feet in length, and two in breadth, so
258 AlIUBING SECRETS.
that the lower edge may be about five feet from the
floor, and coyer it with white Italian gauze, varnished
with gum copal. Provide several frames of the same
size as the aperture, covered with the same kind of
gauze, and deUneate upon the gauze different figures,
8uch as landscapes and buildings, analogous to the
scenes which you intend to exhibit by means of small
figures representing men and animals.
These figures are formed of pasteboard, and their
different parts are made moveable according to the
effect intended to be produced by their shadows, when
moved backwards and forwards behind the frames, and
at a small distance from them. To make them act
with more facility, small wires fixed to their moveable
parts, are bent backwards, and made to terminate in
rings, through which the fingers of the hand ai:e put,
while the figure is supported by the left, by means of
another iron wire. In this manner they may be made
to advance or recede, and to gesticulate, without the
spectators observing the mechanism by which they are
moved ; and, as the shadow of these figures is not ob-
served on the paintings till they are opposite those
parts which are not strongly shaded, they may thus be
concealed, and made to appear at the proper moments,
and others may be occasionally substituted in their
stead.
It is necessary, when the figures are made to act, to
keep up a sort of dialogue, suited to their gestures^
and even to imitate the noise occasioned by different
circumstances. The paintings must be illuminated from
behind, by means of a reverberating lamp, placed op-
posite to the centre of the painting, and distant from
it about four or five feet.
Various amusing scenes may be represented fn this
manner, by employing small figures of men and ani-
mals, and making them move in as natural a way as
possible, which will depend on the address and practice
of the person who exhibits them.
AMUSING S£CR£1«. 259
To direct a swarm of Bees at pleasure.
It is well known that the female bee is the queen of
the hive, and that the fate of the whole swarm depends,
in some measure, upon her alone. The distinguishing
characters of this mother bee are, that she has very
short wings. It is difficult for her to fly^ and therefore
she seldom goes abroad, except when she quits the
hive for a new colony. On that occasion, the bees,
like faithful subjects, follow her to whatever place she
may have chosen, and for this reason, if a person, can
get possession of the queen bee, he is sure of being
able to direct the swarm at his pleasure.
In that case, nothing is necessary but to confine her
by means of a hair, or a very fine thread of silk, made
gently 'fast around her corslet ; the bees, attentive to
all her actions, will surround her, go backwards and
forwards, stop and seem obedient to the will of him
who commands the mother bee, by merely following
the movements of their queen.
This was the charm, or rather the secret, by which
Mr. Wildman, who had studied the instinct of bees,
and who thus took advantage of their attachment for
their queen, was able to make a swarm pas^ from one
hive to another at pleasure. Having full confidence in
the success of his experiments, he presented himself
one day to the Society of Arts, with three swarms of
bees which he brought along with him, partly on his
face and shoulders, and partly in his pockets. He
placed the hives to which these swarms belonged in an
outer apartment, and on blowing a whistle they all im-
mediately quitted him, and returned to their hives ; but
on blowing his whistle a second time, they returned to
occupy their former place on the person^ and in the
pockets of their master. This exercise wa« repeated
several times, to the great astonishment of the society,
and without any of the spectators being injured.
Xhese astonishing experiments, the secret cause of
which we have explained, were repeated some years
agOy with equal success^ before the Academy of
260 AMUSING SECRETS.
Sciences at Paris, by Mr. Wildman, who explained to
the French Academicians the theory and practice of
his wonderful art.
A Powder which inflames when exposed to the Air.
Put three ounces of rock alum and one ounce of
honey, or sugar, into a new, glazed earthen dish, ca-
pable of standing a strong heat, and keep the mixture
over the fire, stirring it continually, till it become very
dry and hard. Then remove it from the fire, and pound
it until it assume the form of a ooarse powder.
Put this powder into a small matrass, or long-necked
bottle, leaving part of the vessel empty, and having
placed it in a crucible, fill up the crucible with fine
sand, and surround it with burning coals.
When the matrass has been kept at a red heat for
about seven or eight minutes, and no more vapour
issues from it, remove it from the fire ; then stop it
with a piece of cork; and, having suffered it to cool,
preserve the mixture in sm«dl bottles well closed.
If you uncork one of these bottles, and let fall on
a bit of paper, or any other very dry substance, a few
grains of this powder, it will first become bluish, then
brown, and will be speedily converted into an ardent
body, so as to burn the paper, or any other combus-
tible substance on which it may have been exposed.
When a few grains of this powder catch fire, on
being thus exposed to the air, they emit a light fiame,
which resembles that of common sulphur when it begins
" to burn ; and they exhale, at the same time, an odour
similar to that produced by the smoke of sulphur*
Fulminating Gold,
Put into a small matrass, resting on a little sand,
one part of fine gold-filings and three parts of aqua-
regia (nitro-muriatic acid). When the filings are
completely dissolved, pour the solution into a glass,
and add to it five or six times the quantity of common
water.
AMUSING SECRETS. 261
Then take spirit of sal ammoniac, or oil of tartar,
and pour it drop by drop into this solution, until the
gold is entirely precepitated to the bottom of the glass ;
decant the supernatant liquor, by inclining the glass,
and having washed it several times in tepid water, dry
it in a very moderate heat, placing it on paper capable
of absorbing all the humidity.
If a grain of this powder, put into a metal spoon, be
exposed to the flame of a taper, as soon as it becomes
sufficiently heated, it will explode with a very loud
report; but it sometimes happens that it pierces the
spoon and forces itself downwards with great violence.
To cut Glass by means oj Heat.
Take a common drinking-glass, not very thick, and
apply to the edge of it a lighted match, until the vio-
lence of the heat produces a crack in it; then move
the match along the crack, following a spiral direction,
and after five or six circumvolutions, the glass will
form a sort of scroll, the parts of which separate when
you invert it ; but which will be re-joined when put
again into its natural position.
This method may be employed to cut glass tubes;
for if a small notch be made with a file in the place
where the tube is to be divided, you may easily make
it split in that place, by applying to it a piece of an-
gular iron made red hot.
To Melt a piece of Money in a Walnut'Shell, without in-
juring the Shell.
Bend any very thin coin, and having put it into the
half of a walnut-shell, place the shell on a little sand,
in order that it may remain steady. Then fill the shell,
with a mixture made of three parts of very dry pounded
nitre, one part of the flowers of sulphur, and a little
saw*dust well sifted.
If you then inflame the mixture, as soon as it has
melted you will see the metal completely fused in the
262 AMUSING SECRETS.
bottom of the shell, under the form of a button, which
will become hard when the burning matter around it
is consumed. The shell employed for the operation
will have sustained very little injury.
Phosphorus,
The name of phosphorus is given to certain bodies
which shine or appear luminous in the dark. Some
kinds of it are natural, and others artificial. The natu-
ral are those which shine without the assistance of art,
such as certain kinds of rotten wood, glow worms,
and almost all fish when they begin to become putrid.
The artificial kinds of phosphorus are those pre-
pared by art, such as Knuckel's phosphorus, (the com-
mon phosphorus of the shops) the sulphuret or the
sulphate of barytes calcined, called Bologna phospho-
rus, &c.
A Liquor which Shines in the Dark*
Takfe a bit of KnuckeVs phosphorus, about the size
of a pea, and having divided it into several portions,
put them into half a glassfuU of very pure water, and
boil it in a small earthen vessel, over a very moderate
fire. Have in readiness a long narrow bottle, with a
well fitted glass stopper, and immerse it, with its mouth
open, into boiling water. On taking it out, empty it
of the water, and immediately pour into it the mixture,
in a state of ebullition ; then put in the stopper, and
cover it with mastic, to prevent the external air from
entering it.
This water will shine in the dark for several months,
even without being touched; and if it be shaken during
dry, warm weather, a kind of brilliant flashes will be
seen to rise through the middle of the water.
Various amusing tricks may be performed with this
phosphorus, by covering the bottle which contains it
with black paper, having words or figures cut out in
it : as you may not only cause dififerent words to ap-
pear, but may even conceal, with one of your fingers,
AMUSING SECRETS. 263
some of the letters which compose them, so as to form
other words, it will seem as if you had the power of
making them appear at pleasure.
- To make Luminous Character^ appear on apiece of Paper ^
or a JVaU, Sfc,
If any characters be traced out with a small bit of
KunckePs phosphorus, they will appear luminous in
the dark. If this experiment be made during warm
weather, the light will be more vivid, and will be the
sooner dissipated, than if performed during cold or
moist weather. By breathing on these characters they
will disappear, but a moment after they will re-appear
of themselves.
A Liquor shut up in a Bottle, which when the Bottle is un*
stopped, becomes Luminous,
Put a little of Kunckel's phosphorus into essence of
cloves, and fill with it a bottje, which must be kept
closely shut : every time the bottle is unstopped, the
whole liquor will appear luminous. This experiment,
as well as the preceding, must be performed in the
dark.
Kunckel's phosphorus may be preserved in a bottle
filled with water ; but it must be put back into the
bottle as soon as it has been used, and care must be
taken not to touch it with the naked fingers, because
it would burn them, and occasion very acute pain;
in short it is impossible to be too careful in handling
this dangerous substance.
Method of speedily delineating all sorts of Plants and
Flowers.
Provide two balls and some printer's ink, then hold-
, ing one of the balls in the left hand, place u]pon it the
leaf or plant, the impression of which you are desi-
rous of obtaining, and taking the other ball, which
must be daubed over with mk, in the right hand, strike
264t ABOJSING SECRETS.
it gently once or twice against the plant, without de-
ranging it. Then carefully remove the leaf or plant,
and putting it between a sheet of paper folded double,
lay it on a table covered with a woollen cloth, and press
it two or three times with a wooden roller, covered with
a handkerchief, or any thing else of the like kind. Af-
ter this process, you will find on each leaf of the paper
an impression of the upper and lower side of the leaf;
which, besides being a perfect resemblance of nature,
will even surpass the most beautiful engravings, espe-
cially if the operation has been performed with dex-
terity.
The Changeable Rose,
Take a common full-blown red rose, and having
thrown a little sulphur finely pounded into a chaffing-
dish with coals, expose the rose to the vapour. By
this process the ro^e will become whitish ; but if it be
afterwards immersed sometime in water, it will resume
its former colour.
The Magic Picture.
Provide a glass similar to those used for miniature
paintings, that is to say, somewhat concave, and an-
other piece of common glass of the same size, and ex-
ceedingly thin. Fill the concave side of the former
with a mixture of hog*s lard and wax melted together ;
then apply the two pieces of glass to each other ex-
actly, that the above composition may be inclosed be-
tween them ; and, having wiped the edges very clean,
cement upon them, with fish glue, a small slip of
swine's bladder. When it is thoroughly dry, clean the
glasses, and apply to the flat side a portrait, or any
other subject at pleasure, and inclose the whole in a
frame, so as to conceal the edges.
If this portrait be exposed to heat, the composition
between the two glasses will dissolve, and become
transparent, and the portrait will be distinctly seen;
but it will disappear when the substance cools. In
this manner it may be made to reappear as often as
you choose.
AMUSING SECRETS. 265'
The Changeable Picture^
Paint upon thin paper, in a slight manner, and with
very light colours, any subject at pleasure, but dis-
posed in such a manner, that by painting the paper
stronger on the other side, it may be entirely disguised.
Then cover the last side with a piece of white paper, to
conceal the second subject, and inclose the whole in a
frame, and even between two pieces of glass.
If you hold this picture between you and the light,
and look through it, a subject will be seen very different
from that which it exhibits when looked at in the usual
manner.
Golden Ink.
As writing, before the invention of printing, was the
only method of transmitting to posterity the works and
discoveries of celebrated men, it became in the four-
teenth and fifteenth centuries, an art much cultivated,
and in which many persons excelled. The manuscripts
of those periods contain writing, the neatness and regu-
larity of which are astonishing. Transcribers were
even acquainted with a method of ornamenting the ini-
tial letters with gold, which they applied in -such a
manner as to preserve all its splendour.
Writing, by the invention of primting, having become
of less importance, soon degenerated, and the secret of
applying gold to paper and parchment, like many other
arts, was at length lost. The Benedictines, however,
re-discovered this secret, and specimens of the process,
and parchment containing writing in gold letters, as
brilliant as those so much admired in the ancient manu-
scripts, have been seem at the Abbey Saint Germain des
Pres, at Paris. This process may be exceedingly use-
ful, and may furnish hints for improving some of the
other arts, which are all connected, and mutually tend
to promote each other*
K
S66 AMI78IN6 fifiCBBTS.
Process translated from the German.
Take a certain quantity of gum arabic, the wlutat
is the best; aild, having reduced it to an impalpable
powder in a brass mortar, dissolve it in strong bran^,
and add to it a little common water, to render it more
liquid. Provide some gold in a shell, which must }^
detached, in order to reduce it to a powder. When
this is done, moisten it with the gummy solution, and
stir the whole with your finger, or with a small hair-
brush ; then leave it at rest for a night, that all the gold
may be better dissolved. If the composition becomes
dry during the night, it must be diluted with more gum
water, in which a little saffron has been infused; but
care must be taken that the gold solution be suffi-
ciently liquid to be employed with the pen. When the
writing is dry, polish it with a dog's tooth.
m
Another Process,
Reduce gum ammoniac to powder, and dissolve it in
water in which gum arabic has been previously tiis-
solved, and to which a little garlic juice has been added. ^
This water will not dissolve the gum so as to form a
transparent fluid ; for the result will be a milky liquor.
With this liquor you must form your letters or orna-
ments, on paper or vellum, by means of .a pen or hair-
brush; then suffer them to dry, and afterwards breathe
on them for some time, till they become some^i^t
moist, and immediately apply a few bits of g(^d leaf cut
to the size of the letters ; press the gold leaf gently
with a ball of cotton, or bit of soft leather, and when
the whole is dry, take a soft brush and ^raw it gently
over the letters, to remove the superfluous g^ding.
The parts which you wish to polish and render bril-
liant, may then be bumish'd with a dog's tooth.
White Inky to wrke on Black Paper,
Take egg-shells, and having carefully washed them,
remove the internal pellicle, and grind them, on a piece
AMUSING SECRETS. 267
"of porphyry. Then put the powder into a small vessel
■ filled with pure water, and when it has settled at the
bottom, decant the water, and dry the powder in the
sun. This powder must be preserved in a bottle.
When you are desirous of using it, put a small quan«
tity of very pure gum ammoniac into distilled vinegar,
and leave it to dissolve during the night ; next morning
the solution will appear exceedingly white, and if you
then strain it through a piece of linen cloth, and add to
it the powder of egg-shells, in sufficient quantity, you
vnll obtain a very white ink.
Red Ink.
Boil four ounces of Brazil wood in two pints of water,
for a quarter of an hour, and having added a little
alum, gum arable, and sugar-candy, suffer the whole to
boil for a quarter of an hour longer. This ink may be
preserved a long time^ and the older it grows, it will
still become redder.
Blue Ink,
Blue ink may be obtained by diluting indigo and
ceruse in gum water.
Yellow Ink.
Take saflron and yellow berries (gramed^ Avignon,)
or gamboge, and dilute them as before, in gum water*
r
Green Ink,
This ink is made by boiling sap-green in water, in
which a little rock alum has been dissolved.
Ink of different Colours, made from the Juice of Violet*.
Dip a camel's hairbrush in any acid, such as diluted
spirit of vitriol, and draw it over a part of the paper.
When the liquor is dry, write on it with a pen dipped
o 2
368 AMUSING SECRETS*
in violet juice, and the writing will immediately appear
of a beautiftil red colour.
If a camel's hair brush, dipped in an alkaline solu-
tion, such as that of salt of wormwood in water, be
drawn over the other part of the paper, by writing on
it when dry with juice of violets, you wiU obtain cha-
racters of a beautiful green colour.
If you write with the juice of violets, and draw a
brush dipped in spirit of hartshorn, or a solution of salt
of wormwood dissolved in water, over another^ you will
have red and green writing.
By exposing this writing to the fire, it will become
yellow.
If you write on paper with an acid, such as lemon-
juice (which is as proper for this purpose as any other)
and then suffer it to dry, the writing will be invisible
till brought near the fire, when it will become as black
as ink. . The juice of onions produces the same ef-
fect.
The older writing of this kind is the more beautiful
the colour becomes ; and in like jnanner the longer the
spirit of vitriol, or solution of salt of wormwood, &c.
has been left to dissolve, before they are used to write
with, the brighter will be the colours.
Tracing Ink.
This name is given to a kind of ink employed for
tracing out figures, and other subjects, intended to he
engraved, as by means of pressure it may be transferred
from paper, and fixed on the white wax with which en-
gravers cover their plates.
To compose this ink tzike gunpowder finely pounded,
and add to it an equal quantity of printer's bladk;
then put the whole into water with a little Roman
vitriol, and stir the mixture, giving it such a consist-
ence that it may be neither too thin nor too thick.
Before the ink is used, shake it well, because the black
is apt to deposit itself at the bottom of the vessel.
1
AMUSING S£CRET8. S69
China, or Indian Inh
China ink, which is employed for small drawings
and plans, may easily be made by the following pro-
cess. Take the kernels of the stones of apricots, and
bum them in such a manner as to reduce them to pow*
der, but without producing flame; which may be done
by wrapping up a small packet of them in a cabbage
leaf, and tying round it a bit of iron wire. Put this
packet into an oven, heated to the same degree as that
required for baking bread, and the kernels will be re-
duced to a sort of charcoal, with which an ink may be
made similar to that brought from China.
Pound this charcoal in a mortar, and reduce it to an
impalpable powder, which must be sifted through a
fine sieve ; then form a pretty thick solution of gum
arabic in water, and, having mixed it with the powder,
grind the whole on a stone, in ^he same manner ;
colourmen grind their colours. Noi^jng i,« iii^«i neces-
sary, but to put the paste into sonte small moulds,
formed of cards, and rubbed over with white wax, to
prevent it adhering to them.
In regard to the smell of the China ink, it arises^
from a little musk which the Chinese add to the gum ^
water, and may easily be imitated. The figures seen
on the sticks of China ink, are the particular marks of
the manufacturers, who, as in all other countries, are
desirous of distinguishing whatever comes from their
hands.
Dr. Lewis thinks, from the information of Father du
Halde, that China ink is composed of nothing but
lamp-black and animal glue. Having boiled a stick
of China ink in several portions of water, in order to
extract all the soluble parts, and having filtered the
different liquors, which he evaporated in a stone vessel,
he found that the liquors had the same odour as glue,
and that they left, after evaporation, a pretty conside-
rable quantity of a tenacious substance, which seemed
to difier in nothing from common glue.
o3
270 AMUSING SECREl*.
Ink Powder.
w
• . Common liquid ink, the method of making which
Ave have already described, is not easily transported
from one place to another; and, besides this inconve-
nience, it is apt to dry in the ink-holder. In bottles,
unless v^ell corked, it becomes decomposed and eva-
porates; and if the bottles happen td break, it may
spoil clothes, or any other articles near.it For the
convenience therefore, of those who^ travel either by
land or by sea, ink powder has been invented, which is
nothing else than the substances employed in the com-
position of common ink, pounded and pulverized ; so
that it can be converted into ink in a moment, by mix-
ing it up with a little wafer. .
■..*■<■
^<« Jtmve Old Writing.
It \b often necessary to consult old charters, titles,
Jeeds, and manuscripts, written many centuries ago,
either to gratify curiosity, or to clear up some import-
ant point in law; but as the writing is sometimes so
much effaced as to be scarcely legible, a Benedictine
invented a liquor, which will make old manuscripts
. appear as fresh as if newly written. The process for
preparing this liquor, which may be easily applied, is
as follows:
Having provided a pot, capable of containing three
quarts of water, take some white onions, freed from the
exterior thick skin, and cut them into small morsels ;
put such a quantity of them into the pot as to occupy
three fourths of it> then fill up the remaining part
with water, and add three gall-nuts well pounded. Boil
the whole for an hour and a half, and throw into th§
mixture about the size of a nut of rock alum. Strain
the mixture through a piece of cloth, squeezing the
onions strongly to express the juice, and preserve the
liquor, which, when cold, will have the appearance of
orgeat.
AMUfiING SECRETS. 271
When you intend to use this liquor, expose it to heat,
which will render it clear ; then dip in it a bit of rag,
and apply it to the writing near the fire, that the liquor
may make a stronger impression, you will have the
pleasure of seeing the characters revived in their full
lustre. If there be only a few words of the writing
effaced, it will be sufficient to heat a little of the liquor
in a silver spoon, and to apply it as above.
Another process, more simple, consists in putting
three or four pounded gall-nuts into a certain quantity
of spirit. of wine; heating the mixture and exposing to
the vapour of it the writing which you wish to revive.
Old papers or parchments, the writing of which can-
not be read, or can be read only with difficulty, may be
immersed also in water in whick copperas has been
dissolved ; if they are then suffered to dry, the copperas
will make the writing re-appear with as much freshness
as if it were new.
To take off the Impression ofciny Drawing,
The impression of any drawing may be taken off, by
placing a piece of glass over the orig-inal, and then
tracing out all the outlines with a bit of soft red chalk;
but as red chalk makes no mark upon glass, it must
first be rubbed over with gum water, to which u little
rinegar has been added ; when the gum is dry it will
be fit for drawing on. Without vinegar, red chalk
would not mark on the gum ; but if you rub the glass
with the white of an egg, instead of gum, there will be
no need of vinegar.
When the drawing has been traced on the glass, if
you applv to it a piece of moistened paper, pressing it
strongly down, and immediately remove it, lest it
should adhere Xo the glass, you will find imprinted on
it the drawing made with the red chalk. By these
means, you will obtain an exact outline of any drawing
or print you wish to copy. This resemblance, how-
ever, will be reversed; and for that reason, to give it
the same appearance as the original, it must be re-
copied.
HH.
373 AMUSING fiECRETt.
To take off the Impression of Old Prints,
Take Venice or Windsor soap, which must be cut
into small pieces, a certain quantity of pQtash, with as
much quick lime, and boil the whole in a pot. Wet
the engraved side of the orint gently with this liquor,
then apply to it a sheet or white paper, and roll it se-
veral times with a roller, in order that the impression
may be complete.
Method of tea^lhig Drwamg to Young Persons.
An artist proposes to teach young persons the ele-
ments of drawito^^ bj making them first practice with
a slate, because it maybe soon cleaned with a wet
cloth, or sponge. This method indeed would save the
expence of paper, and afford the pupils an opportunity
of easily correctin^j their faults, without being obliged
to begin their drawing again entirely. But it is more
arjvantageous to cuijiit^y, instead of a slate, a piece of
Bohemian glasti which might be made rough on one
side, by rubbiuj^ it with a pumice stone, or a flat bit of
free stone, or Une sand well moistened. Whatever
figures have been drawn on this glass, may be effaced
by a wfvj cloth, in the same manner as from a slate ;
and besides this advantage, as the glass is transparent,
cofrect copies may be placed below it, which the
scholars ovic^ht to follow till their hand is properly
formed. What is here said of drawing may be applied
also to writing.
To construct a Lantern which mil enable a Person to Read
by night at a great Distance.
Make a lantern of a cylindric form, or shaped like a
small cask placed lengthwise, so that its axis may be
horizontal, and fix in one end of it a parabolic or
spheric mirror, so thati»its focus may fall about the
middle of the axis of the cylinder. If a small lamp or
taper be placed in this focus, the light passing through
AMUSING SECRETS. 273
the other end, will be reflected to a great distance^
and will be so bright, that very small letters on a re-
mote object may be read, by looking at them with a
good telescope. Those who see this light, if they be
in the direction of the axis of the lantern, will think
they see a large fire.
To take off Impressions in Plaster of Paris or Sulphur,
As' curious people, who cannot purchase the originals,
are often desirous of obtaining impressions of medals,
engraved stones, and other valuable articles preserved
in cabinets, they may easily be procured, and at a very
small expence. The whole process consists in a very
simple operation, which will give a striking resemblance
of the object, so as to exhibit all its parts with the
greatest truth.
When you intend to take off an impression in plaster,
that which has been pulverised and sifted through #
piece of very fine silk must be employed. First rub
over the medal, or engraved stone, very softly with
oil, and having wiped it with cotton, surround the edge
of it with a bit of thin lead. Mix up the sifted plas-
ter with water, and stir it gently, to prevent it throw-
ing up air bubbles; then throw it over the medals, and
suffer it to harden and dry. It may then easily be de-
tached, and will form a mould, strongly marked, by
means of which you may take off impressions in relief,
either in plaster or sulphur. — Observe, before these
moulds are used, they must be impregnated with oil.
The process for melted sulphur is the same as for
plaster ; but it is to be observed, that when the model
is of marble, old lard ought to be employed in pre-
ference to oil, because the latter, by penetrating through
the pores of the marble, would stain it.
Baits for Catching Fish,
In order to attract fish when angling, baits made of
various kinds of grain, such as wheat, barley, oats, or
274 AMUSING SECBim.
boiled beans, mixed with aromatic herbs, and pounded
with earth, may be employed. Fish are wonderfully
attracted by strong smoiling' substances, as camphor,
assafoetida, Sec. They seem to have a great fononess
for a paste made of crusts of btead, honey, and aisa-
foetida. It is said also, that they approach colooied
objects through curiosity.
Some people tie a bit of scarlet rag to the hook, and
rub it over with petroleum; and others highly extol
heron oil. To obtain the latter, the flesh of the heron
is cut small, and pounded in a mortar ; it is then put
into a long necked bottle, closely corked, and pre-
served for two or three weeks in a warm temperature ;
the flesh, by putrefying, is converted into a substance
that approaches near to oil, which is mixed up with
honey, bread, and a little milk. Most fish, and parti-
cularly carp, are said to be very fond of this bait.
Artificial insects are much used also for catching
fish, especially trout;. they are made of different co-
lours, according to the hours of the day, in order that
* they may imitate the natural objects which appear at
these different periods.
Those who fish in fresh water, employ cheese some-
times as a bait, and prefer that which emjts the strong-
est smell. The putrid livers and flesh of animals of
every kind are likewise used.
Small, long, slender worms, of a white or pale yellow
colour, with a red head, contained in small cells found
in the roots of the water iris, are said to be excellent
bait for trout, tench^ carp, and various other kinds of
fish.
Earth worms, as well as those engendered in meat^
are of great service.
To procure the latter, and almost at every season,
a dead cat, or bird of prey, must be exposed to the
flies, and when the worms become very lively, it ought
to be buried in moist earth, as much sheltered from
the frost as possible. The worms may be taken out as
they are wanted. As these worms are metamorphosed
into flies towards the month of March, recourse must
the^ be had to other animals of the like kind.
idlfllUNG 8ECRIZES. !S75
To produce Variety in the Colours of Flowers,
Variety is generally produced in flowers by sowing,
in the same bed, seeds collected from different indi«
viduals ; and there is reason to think that this variety
in colour arises from the farina of the differently co-
loured flowers, which mutually fecundate each other.
This conjecture is supported by experience ; for it is
found, that if flowers of the same kind, but different
in colour, that is, some red and others yellow, flower
together, the seeds arising from them produce red,
yellow, and orange flowers, ^nd even some diversified
with red and yellow. It is certain also that the varie-
gations of flowers are more singular, according as the
variety of colours contrasted together in the same bed,
is greater ; that by planting together in the same pot
yellow and white ranunculuses, the seed resulting from
them will produce sulphur-coloured ranunculuses;
and that aurora-coloured ones may be >obtained, in
like manner, by a similar process, with yellow and red
ranunculuses.
It may easily be proved by experiment, that this
phenomenon arises only from the influence of the fa-
rina; because, when these flowers are planted sepa-
rately, and at a distance from each other, they pro-
duce only the same colours.
To obtain Double Flowers,
t
The more petals a flower has, it becomes the fuller
and more beautiful. Flowers sometimes are converted
into double ones by accident, but there are some which
are only very little so, as may be observed among car-
nations. There is, however, an artificial method of
making them become double, which is, to transplant
them several times the first year, as in spring and au-
tumn, without suffering them to flower. By following
this method for two years consecutively, single carna-
tions may sometimes be converted into double ones.
^6 AMUSING SECRETS.
To obtain Flower 9 of different Colours on the same Stem,
Scoop out the pith from a small twig of elder, and
having split it lengthwise, fill each of the parts with
geeds that produce flowers of different colours. Sur-
round them with earth, and then tying the two bits of
wood, plant the whole in a pot filled with earth pro-
perly prepared.
The stems of the different flowers will thus be so
incorporated, as to exhibit to the eye only one stem,
throwing out branches covered with flowers analogous
to the seed which produced them.
By selecting the seeds of plants which germinate at
the same period, and which are nearly similar in regard
to the texture of their stems, an intelligent florist may
obtain artificial plants exceedingly curious.
THE END.
Plwnmer and Brtwis, Printers, Love Lane, Little JEastchemp,
ill
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